So fun fact everyone. SCOTUS has already ruled on something very similar when it came to Jehovah's Witnesses being forced to stand for the pledge. Guess what they ruled. Go on guess.
Ironbite-the NFL may be violating the Constitution.
Actually, when it was specifically about Jehovah's Witnesses as a religious matter, the Court ruled against them, holding that Free Exercise did not prevent a school from punishing a student who refused to stand for the Pledge. Their overturning of that ruling (though it was again in a case involving a Jehovah's Witness) rested on the Free Expression clause, holding that anyone had the right, as a matter of speech, to refuse to stand for the Pledge or salute the flag or what have you. (Which is a stronger holding, to be sure, but nonetheless it is a speech, rather than religion, matter, and religious rights are still, technically, circumscribed and those refusing to stand for the Pledge are invoking their Free Expression rather than their Free Exercise rights in doing so.)
But, more saliently, a school is, at some level, an extension of the government (if nothing else, students are bound to be there, whether under threat of legal punishment for themselves or their legal guardians), which is bound to act in accordance with the Constitution, which forbids them from acting in a manner to abridge the rights of free exercise and free expression. The NFL, by contrast, is, ultimately, a private institution. Nothing governmental compels players to play for a team in the NFL. Nothing legal bars a player truly opposed to this from attempting to set up his own competition; if the NFL attempted to prevent that, it would be acting in a monopolistic manner forbidden by antitrust law. Further, some owners (like the Jets' owner) have said that they will personally pay any fines imposed on their players or their team should a player act in contravention of this ban and not compel players to abide by it. (Apparently one version of this proposal had the offending team suffer a 15-yard penalty, which would have been a very different matter.)
I would even note that there is another professional competition, contested under similar (though not identical) rules to those employed in the NFL, conducted in North America, during roughly the same time frame as the NFL, and I imagine they wouldn't mind an influx of talent.