One small problem:
Define "Health care" as a "right".
Does this "right" include exorbitant measures to extend life?
No. Healthcare is rationed in all current societies. In some societies the decision between whether someone receives treatment or not depends on whether they have insurance or money. In other societies, the decisions are based on other factors.
Would it include plastic surgery (you know, for self-esteem reasons)? Does this "right" diminish with age, since old people getting a scarce resource (e.g. organ transplants) wouldn't see nearly the benefit from it that a younger patient would? I could go on, but you get the point.
These are all questions of health rationing, normally decided by a government organisation (such as NICE in the U.K.) who sit down to thrash out the priorities. Some choices are easy - i.e. $50000 for treatment and ITU care for a 20 year who has had a heart attack due to a congenital heart problem versus $50000 for a series of cosmetic operations. Obviously, there are much harder decisions - but that doesn't mean one shouldn't try to make them, and revert to a system where these decisions are based on the ability to pay.
Obviously there has to be limits on what should go into health care. That said, it's one thing to set those limits impersonally. It's another to see these limits in action when it's your spouse, parent, or child that runs up against them.
Generally children are not ones that are limited against, but yes, if your father is in need of a liver transplant because of alcoholic liver disease, the next organ is more likely going to go to the 25 year old with the autoimmune cirrhosis in need of the liver.
BTW - two things:
1) since when does a right include automatic access to another's labor? Speech, privacy, and all the fun rights listed in the US Constitution don't require another's labor, time, or money.
Really? Freedom of speech requires a fully functioning democratic government that supports freedom of speech - forcibly if necessary, with the full backing of economic stability and peace. Privacy - same again.
Your "right" to health care does. Why is that?
2) If I choose not to exercise an enumerated right (again, c.f. US Constitution), it costs me nothing. If I choose not to exercise this "right" to health care, I still have to pay for it. What the hell?
That is true, but it is unlikely that you will never wish to make use of health care - but I'm guessing you mean that in that scenario you will pay for it yourself. If you are in a position to pay for it yourself, you are likely a high earner. An argument can be made that your wage is dependent on a number of people with a lesser wage to support your position, and it is in your interest that these people are able to receive health care.