I was going to get to this later, but a couple of people have brought it up, so let's get it out of the way now: what about the sewers? Or vaccination? Or the many other public efforts that have made people well?
Those efforts are justified on a completely different moral logic than something like single payer. Because of the way that disease spreads, things like sewers and vaccinations are a genuine public good. That is, they have significant positive externalities from which your neighbours cannot be excluded. If I get vaccinated, that lowers your chance of disease, even if you don't get a vaccination. Likewise, if I treat my sewage, you become less likely to get cholera, even if you don't treat yours.
As long-time readers of my old blog know, I'm pretty harsh on people who don't vaccinate (or use sewers, either, though that one hasn't really come up). The problem with vaccines is that they've been too effective; effective enough that parents are (rightly) more worried about a small risk of side effects from a measles or polio shot, than they are about the risk of blindness, heart disease, or paralysis. That tempts them to free ride on other parents who do vaccinate.
As an individual, that's the smart strategy, but socially it's disastrous, since it destroys the compact by which we keep infectious disease at bay. Also, once there's a new reservoir of unvaccinated kids, their free ride becomes not that free.
But treating infectious disease to keep it from spreading is in a different moral category from a universal health care system. Curing my asthma will not protect the people across the street, or America, from danger.