There's a reason the Founding Fathers enumerated very few "inalienable" rights, because there are only a few, no matter how fundamental they are to our society, and civilization in general.
More specifically, no material right can be inalienable by definition. This is something very few people seem to understand and yet the foundation of "liberal" policy assumes the very opposite. You cannot have an inalienable right to a finite resource, because it's always possible for there to simply not be enough of that resource.
So, do you have a right to have access to health care? Sure.
Reasonably priced health care? Yes, whenever possible. You have the right in general not to be extorted. Therefore you have the right to be charged a fair amount for services and products. What's fair? That's another can of worms, but let's move on.
Free health care? Absolutely not. This is not a right and cannot be a right, even a non-inalienable right. It's literally impossible to provide it so how could it be considered a "right", especially given the extremely vague nature of the concept of "health care", which today is considered to comprise a much broader range of products and services than what is literally needed to cure sickness (as opposed to maintain health... it's an important distinction).
If you do have some form of health care that you do not pay for directly, it's a privilege. It's arguable that to grant everyone this privilege is a good idea and an attainable goal (although I don't think it is), but we are not talking about rights.
We are decades too far into this debate to still have this much trouble defining our terms.