For the record, this article is not motivated in any way to take a specific moral or political stance on what have been extremely controversial issues. The intent is to clarify what the law is in a social media atmosphere that has made it hard to understand. A lot has been stated regarding what NFL players are and are not allowed to do, what the government is and is not allowed to do and what NFL owners are and are not allowed to do. Whether one side or the other is "right" (or left) is for a different kind of article (there are plenty, let's not add another one), but let's clear up the legalities of the situation now.
The first amendment is a complex beast. For starters, it is not limitless. You can still get into legal trouble for endangering the safety of others with false statements (i.e. yelling fire in a theatre when there is no fire); for harrassing or threatening another person; for stating things about someone else to others that you know is not true (i.e. libel, slander); generally speaking, it is better to stay away from talk that suggests you are actively trying to overthrow the government.
Other than that, most everything else--speech, expressions, acts--that do not "disrupt the peace" is fair game. To be clear, Neo-nazis can march down a street with tiki torches and chant things I won't even repeat. NFL players absolutely have a constitutional right to kneel during the national anthem, as does anyone else. This does not mean, however, they have a right to keep their jobs. This is an entirely different legal concept all together.
The constitution protects citizens from the government; it does not protect them from the private workplace, nor how other people will react to a citizen's statement or act. When I say that players have the constitutional right to kneel, I mean they are legally protected from the government--police officers cannot suddenly run out onto the sideline and handcuff anyone that isn't standing.
Depending on the employment contract, however, a boss can still fire an employee for excercising freedom of speech. Both an employee's constitutional right to go into their boss's office to tell him he is a raging moron while burning an American flag and the boss's contractual legal right to terminate the employee for doing so can both be correct statements (though, the employee could potentially be arressted for causing a fire hazard...). Depending on the NFL player's contract, an owner could potentially release that player for kneeling (for financial and other reasons, however, whether they would or not is an entirely different conversation).
When one of the Robertsons from the Duck Dynasty television program came out and said negative things about gay people, he was exercising his freedom of speech. No one could arrest him for saying that. Under his television contract, however, the network had the legal right to pull the show.
Freedom of speech does not protect a person or business from how other private citizens will react. People upset with what is happening in the NFL have every right to boycott it or support what the players are doing or otherwise so long as it is peaceful. When a local store recently made it very public what their political stance and beliefs were, they were exercising their freedom of speech. The local reaction to the store's statements forced the store to close, a private citizens' reaction to a public, constitutional right.
Other than some exceptions stated above, you can say whatever you want to say without government interference. That does not free you from consequences you might encounter in your private life with friends, employers, businesses or entities other than the government.
On a semi-related sidenote, whether Colin Kaepernick has a valid grievance that NFL owners colluded not to hire him likely depends on what the collective bargaining agreement between the players' union and NFL says--again, it is not a constitutional issue of whether they are violating his freedom of speech. If the agreement forbids such owner collusion, he likely has a claim. He would still have to prove the collusion existed and owners did not hire him for that reason, as opposed to other plausible reasons.