They don't just get paid for game time. If a player never plays in a game, he still gets paid. NFL players have to attend practice, team meetings, workout sessions, ect. When they are in uniform and on the field on game-day, they are representing their team and employer.
I think it is ultimately up to the team owner, if he wants to allow it, then I think it is OK. He will lose a lot of revenue because of it, but it is within his rights to allow it.
Otherwise, employees typically don't have a "right" to engage in protests while "on the job". Do you see Best Buy employees protesting while at work? Do you see police officers or government employees protesting while at work? No.