There is actually a congresscritter who wants to make the following amendment to the constitution:
"all persons shall enjoy the right to health care of equal high quality."
U.S. Rep. Pete Stark, D-Fremont
As others have pointed out, there are other even more basic requirements than health care. Why not an amendment:
"all persons shall enjoy the right to food, clothing, and shelter of equal high quality."
What about legal services? The legal system is one of the functions of government, but while poor people charged with a crime can get a court-appointed lawyer, I'm pretty sure most of them aren't the Johnny Cochran's of their class. Shouldn't we have an amendment mandating legal representation of equal high quality? We can put price caps on it and everything, and if we mess it up and lose our brightest lawyers at least is wouldn't be as bad as losing our brightest doctors.
All this may seem like I'm against universal health care. Actually, I wouldn't have an objection to it as long as the following criteria are met:
1. Preventative services (vaccinations, prenatal care, etc) are funded before anything else.
2. Medical expenses for things caused by lifestyle choices (obesity, drunk driving, smoking) are paid by the patient.
3. Medical services are preferentially given to those with a greater chance of contributing to society -- young people get organ transplants before old people, with criminals on death row at the back of the line.
4. Some mechanism for restricting "vanity" and unnecessary care (facelifts and hypochondria are kept out).
5. People who use more health care pay a greater percentage of their take-home pay for insurance.
There are probably other criteria as well, but that would be sufficient to start with.