Your private employer may restrict your speech pretty much any way it likes, with a few important exceptions such as the preferential application of restrictions based on race, religion, gender, orientation, etc. Obviously you can't do that.
Kaepernick isn't freely speaking because the Constitution allows him to, he's speaking freely because NFL commish Roger Godell allows him to.
All Godell has to do is make a rule this off-season stating "no political speech while in uniform or while officially representing the NFL or its franchises. What you do on your own time is your own business".
I think that would be a good rule, otherwise we're going to start seeing every disgruntled champion-of-causes making in-game political statements in all sorts of ways and it's going to turn off a lot of fans.
Besides, Kaepernick is only drawing attention to himself, not his cause.
Here's Kaepernick, exercising the coaches right to sit him on the bench.