The NFL has no rules for the National Anthem. Players have sat for many years. They now recommend staying in the locker room if you choose to not stand.Is it their right? I'm not so sure. It's all whether their contracts allow it.
The leagues are private companies and, like most, can restrict your speech at work or on employee property. So, they do surrender 1A protections in a sense, at work on work property. It depends on if their contracts allow such behavior, or the league allows it.
I suspect it violates terms of both, but I also suspect the league has turned a blind eye to the behavior.
Of course, NFL owners are the bosses. They can hand down punishment if they want.
There is Federal Law that defines etiquette, but I believe it is not mandatory.