If a doctor ever asked me to sign something saying to keep my opinions to myself, I would simply not use that doctor. If they were good, they shouldn't worry about what people will say about them.
While this should be the case, I don't like this argument because it reeks of the "what do you have to hide?" argument used against privacy protections. The problem isn't the good doctors, it's the bad doctors.
I'd love to see an example of how this gets used by doctors in the wild, to make it easier for me to protect myself against such absurdity.
That said, if my PCP asked and I had pressing medical concerns, I'd despise making the choice of 'free speech' versus 'need medical care', but I'd certainly choose the medical care.
This is why pre-nups become unenforceable once someone gives an ultimatum (ie. Sign this prenup or I won't marry you) and why both parties will have lawyers.