Personally I would lean towards placing healthcare in the same category of critical public service that the gov't should provide, along with fire departments, police, etc. The thing in common is that they all deal with life and death issues. If one doesn't think that the government should be providing healthcare services, then it's easy to see how one would think that we should all be hiring private security forces and fire brigades. The only reason that no one (or very few people) think this, is because police and fire departments became public services a long time ago. If we had gone with a gov't single payer healthcare option back in the 40s or 50s, like most every other first world country, we would probably not think anything of it now, anymore then we do fire departments and police departments. We would just accept it as normal.
An argument could be made that healthcare workers require much more training/education etc, and that that is expensive and/or that because of that those people deserve to be paid well. I won't disagree. So if we have public healthcare then lets make sure they get paid appropriately for what they do.