There is no nation-wide vaccine mandate. The fact that you can't do certain things or work certain places without being vaccinated does not take away your right to choose. You do not have a Constitutional right to fly or to go to a restaurant. Your employer already has broad latitude to both require and forbid you from certain things. In most states they can fire you for a good reason, a bad reason, or no reason at all, as long as it isn't an illegal reason. They can fire you for wearing something with the name of a political candidate they don't like.

False impressions about what free speech and constitutionally protected liberties actually are lead to all sorts of silly ideas.