The actual legal issue here is not the one we see argued over in social media or even over the airwaves. Is "speech" involved? Yes. Is "Constitutionally protected" speech involved? No. So, all the folks you read or hear saying it is the latter are wrong. The 1st Amendment has no application here. The people who say it does are either uneducated or, worse, they already know this but also know their audience is dumb so they make the claim anyway. Which is worse? You can decide that.
What is involved are employees' duties and rights while on the job. There is a large body of law called "Employer-Employee" that has already sorted this kind of stuff out. In short, employers may control the speech of their employees, generally. Employees may be legally fired for breaching an employers dictates on speech while on the job.
So, can pro football players protest? Of course they can. The have the same rights as anyone else. But the issue here is - can they do it while on the job, in uniform, at the workplace? The answer is no, not if the employers says they can't. Or, perhaps more pragmatically, can they be penalized, even fired/cut for violating their employers' rules? The answer again is of course they can. Attempting to wrap themselves in the Constitution (an ironic move given that the protesters hate it and what it stands for) will not protect them.
The examples are easy to come by. Can an order taker at McDonalds cuss out customers? No (well they can, I guess, but they can be legally fired for it). Can Home Depot workers wear "F*ck Home Depot" t-shirts on the job? Again, no. Can Walmart greeters burn effigies of Hillary in front of the store? It's a form of speech, right? The law says it is indeed speech but the law also says they can be fired for it. On top of that, employment in most states is "at will." Which means employees can be fired for almost any reason.
Is the NFL different? The NFL does have a CBA and standard player contract. But as far as I recall, neither carve an exception for speech. Indeed, we have already seen the league and Commissioner's Office establish precedent in this area by cracking down on many different forms of player speech. Jim MacMahon's headband. Messages written on shoes. Even a patch to honor 9-11 was prohibited. Kneeling for the anthem/flag before an NFL game is no different.