Where I think this case is different is that this man worked for the government. I think one should have to say something pretty extreme to get shut up by the feds. (Remember the Dept. of Agriculture employee who was fired for being "reverse racist", except that her comments were out of context? That's why the gov't shouldn't fire people for words.)
The hospital in your example would have a hard time doing anything about your anti-vax guy if he didn't do it at work - and if he did it at work, their first move has to be to ask him not to do it at work. Anything he does in his private time is protected. Even stuff he does in his public time is protected - Target, for example, has a league of pharmacists who refuse to dispense birth control pills and they're constitutionally protected.
As for the Target employees, I think they should be able to be fired. Should a Jain be able to work at McDonald's and refuse to serve anything but fries?