How can you logically have a right to the services of someone else? Health care is a product - doctors, hospitals, medicine, etc. These must all be produced by people.

You can't divine rights out of some fundamental logic. The only ones who can decide what rights humans have are the humans themselves, and then we structure our society to make sure those rights aren't violated.

Didn't you say this earlier, though?

The purpose of government is to protect our rights.

It sounds like you also think "you logically have a right to the services of someone else", otherwise who's doing the protection?

"You can't divine rights out of some fundamental logic" - of course you can. This is the role of philosophy.

Good luck with that.

Maybe I should have been clearer: Just because you say it's derivable "from logic" doesn't mean it is. No logic system can do anything without axioms, and the adoption of those axioms is an arbitrary choice.

Axioms are not arbitrary - they simply are. An axiom is a recognition of the nature of reality.