When I think of a right (e.g., I have the right to life, liberty and the pursuit of happiness), I think of the right not to be disturbed in the exercise it. A right in this sense is most emphatically not the right to be given a good or service; that would be an entitlement.
The right to life means the government and your fellow citizens may not kill you (excepting only the punishment of death for serious crimes), not that the government guarantees you that it will help keep you alive. Ditto the right to liberty: you have the right not to be jailed, or enslaved. And pursuit of happiness never meant a promise that things would go well and you would be happy, just that you got to choose how to attempt to achieve happiness rather than being forced into a particular career path or hobby or family.
When people talk about health care as a right, they lose me; I get stuck thinking about whether I agree that it's a right, and how I define rights, and fail to follow the rest of their argument. I've read justifications about how you can't have your life without health, and what good is liberty or happiness without health, but they don't convince me. Health care doesn't fit into my general definition for the kind of right we enforce against our governments (be they private, like families, or public, like the State).
Health care is a duty parents owe their children, and I can't extrapolate from that to governments owing this duty to their citizens; we're not in a parental relationship that way, and I wouldn't want to be.