Is it right for a Jewish Orthodox organization, outside of Israel, to demand that a non-Jewish organization accommodate their religious requirements? Should a boys' basketball team forcefully request a non-Jewish state-sponsored basketball league to change the playing schedule to accommodate their need not to play on Shabbat?

A timely question!  It bears similarity to a recent story of the Beren Academy in Houston, Texas, a Jewish day school which made it to the basketball semifinals in the Texas Association of Private and Parochial Schools league.  The game was scheduled for Shabbat, and while the team was prepared to forfeit rather than play on Shabbat, a lawsuit filed by some parents, together with widespread pressure and support for this team from around the country, compelled TAPPS to reschedule the semifinals and finals to make it possible for the team to play not on Shabbat.  See coverage of the whole story here: http://sports.yahoo.com/blogs/highschool-prep-rally/orthodox-jewish-school-beren-academy-play-state-semis-124507168.html.  As opposed to your question, TAPPS is not a state-sponsored organization, and it was not the team per se (but rather some parents) who made this request.  It is especially noteworthy that many teams have religious attachments in other faiths, and part of the argument made by the ADL in its letter, and by others, was simply that TAPPS should be as sensitive to the religious practices of its Jewish players as it is to its Christian players.

To your question, then: It's hard to say from a halakhic perspective whether it is right or wrong for an Orthodox organization to make this demand of a non-Jewish organization.  The principle of dina di-malkhuta dina, that the law of the land is the law, demands (in most cases) following laws as they are legislated, but it does not preclude advocating for laws to be changed.  Why should a Jewish organization not request the right to participate in the activities of the country in a way that accommodates its religious needs?  This is especially true considering the sources, Biblical (Jeremiah 29:7 - "seek the peace and prosperity of the city to which I have carried you into exile. Pray to the LORD for it, because if it prospers, you too will prosper.”) and on (Pirkei Avot 3:2 - "Rabbi Chanina the deputy [High] Priest said, pray for the welfare of the monarchy, for if not for its fear, a person would swallow his fellow alive."), which advocate praying for the welfare of the lands in which we live and supporting and participating in government!

From a Jewish values perspective, turning the question on its head, we recognize that as Jews we are enjoined throughout the Bible to be sensitive to the stranger among us.  This is a value when Jews are in power in their own land, and perhaps it points to advocating for that same treatment of us as "strangers" in another land.  It seems to me there is nothing wrong with making this request strongly - it shows a commitment to Jewish law, and a desire to fully participate in American life.  Advocating forcefully for that opportunity compromises neither.  Of course, Orthodoxy demands prioritizing the former over the latter when they are not reconcilable (as the Beren students were committed to all along, to their credit).

The broader question of the impact of our treatment under government, and how much we seek equality, and its costs, is treated wonderfully in an article by Marc Stern entitled, "Egalitarianism and Halakha: An Introduction", which appears in the Orthodox Forum Volume from the 2001 Conference entitled "Formulating Responses in an Egalitarian Age".  The article frames these questions beautifully in terms of the cost-benefit analysis internally to the Jewish community in regards to making these demands, and is a worthwhile read in light of this question.

As a Conservative Jew, observant of Shabbat and Festivals, I would support a Jewish organization’s right not to play on Shabbat or Festivals.  The questions of whether it is right to demand that the organization accommodate their religious requirements is a completely different question.  My prayer is that in the coming years, organizations will be more respectful of the different religious holidays and sabbaths of their members.  As such, this question would not be necessary, as the organization would first ask the teams what their availability was, rather than the other way around.
The example given was of the Texas Beren Academy, who were asked to play on Shabbat and were not given an alternative playoff time until parents sued in federal court for a restraining order.   At which point, TAPPS, the Texas Association of Private and Parochial Schools, gave an alternative time for their semi-final game.  According to the TAPPS website, the boys made it to the final game and then lost 42-46.  Win or lose, a great achievement!
In my personal opinion, I believe that Jewish organizations have the right to request for accommodations, but I do not necessarily feel that we have the right to demand.  I think it was appropriate to have a media campaign to ask for the right to play, but I am not convinced a federal lawsuit was necessary.  At the same time, I am incredibly disappointed with TAPPS, that they could not accommodate the team without outside influence, nor did they accept the team’s initial appeal.  It is sad that the national media campaign did not change their mind, but the threat of a lawsuit did.
The other issue here is the question of contract law.  When a school signs up to join an organization like TAPPS, they are joining a private organization with its own rules.  By joining the association, they are accepting the rules, even if they are not fair.  They can lobby to change those rules, but as a private organization, and not having a J.D., I do not know if a private organization is required to satisfy the religious needs of all its members.  TAPPS was formed originally for Christian schools, partially to avoid games on their (Christian) Sabbath, Sunday.
I would hope that an organization like TAPPS would accommodate an organization like Beren, but I do not see that they are legally required to do so.  Beren is obligated to ask, as according to our tradition, while they might technically be allowed to play on the Sabbath, they would not be able to drive to the arena, buy food, properly dry off after washing, etc., or do any number of other things that are required for a championship game.  I do not think it is wrong for them to demand, but neither do I see it is right.

Three links related to this discussion:
The controversy: http://www.nytimes.com/2012/03/01/sports/basketball/texas-association-criticized-for-denying-jewish-school-a-title-shot.html
The initial resolution: http://www.foxnews.com/us/2012/03/02/jewish-basketball-team-wins-playoff-game-after-rescheduling/  
Final score:
http://www.bracketmaker.com/tmenu.cfm?tid=402851&tclass=Basketball%202A%20Boys

This question likely stems from the recent news about Beren Academy, an Orthodox Day School in Texas, whose basketball team earned a place in the semifinals of the Texas Association of Private and Parochial Schools league. The school had been told that any games which fell during Shabbat would not be rescheduled. When they earned the slot in the semifinals, a game set for 9pm on Friday, the school and the players accepted the decision (meaning they would forfeit a game scheduled for Shabbat), but parents sued and eventually overturned the association's decision. The game was rescheduled and the team played (unfortunately losing the game).

The story does not precisely match the question. The appeal was not made by the school, but by individual parents. The individual schools in the league were, according to reports, very willing to accommodate the Shabbat observance of the school since many of them are also religiously based, though the association decided that it would stick to its original rules nonetheless. Indeed the tournament was scheduled with an eye to avoiding Sunday games in deference to those Christian teams that would not play on their Sabbath. It is also worth noting that the Association of Private and Parochial Schools is not state-sponsored, but is a private organization.

Nonetheless, the more generic question remains. Should Jews living in a secular society demand that the outside world accommodate their religious requirements? The answer depends.

According to Federal Equal Opportunity laws there are many instances in which employers and others providing public accommodations must consider the religious needs of their employees or their users, Jews included. An extreme case was Goldman v. Weinberger (1986) when the Supreme Court ruled that a Jewish Air Force officer could be denied the right to wear a kippah. The Court ruled that the Free Exercise clause of the Constitution applied less strictly to the military than to others. Congress, in turn, amended the law to read that “a member of the armed forces may wear an item of religious apparel while wearing the uniform of the member’s armed force.” In America we certainly have the opportunity to ask, even if we will not always prevail.

The question focuses on the limits of participating in a voluntary, secular organization. The key word, of course, is voluntary. The association has the right to set their own rules and to stand by them. They also have the right to negotiate, consider those whom they serve, and to make appropriate modifications. In the case of Beren Academy, the individual schools in the tournament were very willing to accommodate the needs of the Jewish school. If conditions had been different, if the request to modify the schedule would have caused unacceptable losses, the association is within its rights to hold the course and refuse to change. The Jewish organization, having known the situation from the first, would need to agree.

It strikes me that there is one more aspect to this question: should a Jewish organization (or individual) in a non-Jewish setting assert its (or his/her) religious particularity and difference in a public dispute? I feel there are two sides to this question. First, we always have the right to advocate for our needs within the society. Too often Jews were told to just be quiet and let it pass, and the effect was to feel that we were less deserving, less entitled to full participation in our society. Conversely, we should be equally willing to stand by our beliefs, to assert that our identity and our values are more important than outside issues, including a basketball game. Like many rabbis I have had parents explain their children's absence from Hebrew School by telling me how important various school or sports events were in the life of their child. I respectfully disagree and wish they would tell the coach or the teacher that their observance of the High Holy Days, or Passover, of Shabbat was a key value in the life of their family and it takes precedence.

Is it right to ask a non-Jewish organization to accommodate your needs? Certainly. Will you always win your case? No. And I hope in that moment you will stand by your Jewish values.