This is a really interesting question and the trouble is that "healthcare" is such a broad subject.
It's easy to argue that the person who is diagnosed with something like childhood leukemia doesn't need their parents to have to choose between bankruptcy and death. If you're caught up in something totally unexpected it kinda sucks to be stuck with a honking great medical bill to make it right afterwards.
That said it's far more reasonable to expect the person who refuses to take any part in living a healthier lifestyle to take some responsibility rather than simply dump all their medical costs onto society. Why bother to eat healthily when you can just wobble into the doctor's office and demand that Someone Else pay for the drugs you need to shed that excess weight, even as you continue to stuff your face with every kind of junk food?
One fundamental problem with anything that's free at the point of use is that some people regard it as priceless and others regard it as worthless. If you don't have to pay anything to see the doctor and don't have to pay anything for a prescription what follows is depressingly predictable - people with a headache go to see the doctor to get a prescription for an aspirin, creating ever-longer waiting times for people who really need to see the doctor.