No, they don't.
The First Amendment is strictly about Congress, not individual citizens. It proscribes CONGRESS from abridging speech.
Why is this so hard to understand?
I find it remarkable that anyone can read the title of this thread and agree that NFL players don’t have Free Speech rights. All Americans have Free Speech rights. Just because the little kerfuffle today has nothing to do with Free Speech doesn’t mean that NFL players don’t have Free Speech rights.