To concede your point to a significant extent:

It's true that one of the non-State options in a free society was available in the real scenario, as you said, namely not accepting the change of conditions, quitting (or accepting being fired, or not getting the job), and then taking to the public square to try to inflict a reputational punishment to the violators.

It's just that the effect of the State as the default enforcer over the real scenario is so large, that the result of the non-State option would be negligible compared to what it would be in a free society attuned to such mechanisms. 

 But relying on the State to either prevent or enforce company mandates is just perpetuating that power imbalance. 

 I think a scenario of non-mandates in either direction (pro- or anti-vaccines), with the state "just" as enforcer of contracts (verbal or written, doesn't matter) would be a huge improvement. I'm more of an anarchist myself, but many libertarians (minarchists, etc) would be happy with such status quo.