You don’t have free speech nor the right to protest while you’re an employee of a company.
Would you be able to keep your job if you protested on company time?
Nonsense.
The National Anthem playing before a game is such a fluky American thing that no one else does; I don't see how this question holds water when comparing to a normal job. In fact, by giving players the option to remain in the locker room, the NFL is implying that being present for the anthem is not a strict work obligation at all--enough to give them the option of skipping it, anyway. It's not like they are ignoring typing emails or filing recipes or whatever minutiae in order to protest. They are simply making a point during a non-obligatory pregame ceremony that is completely pointless.
This whole episode could have been avoided if 1) the league would grow a set and stand by its players and 2) these offended American patriots would take a second to educate themselves, try to understand the world from the point of view of people who are murdered and brutalized by police, and consider that maybe all isn't Holy and Good in this clusterf*** of a country.