I believe that there should be a safety net. People should get a reasonable level of healthcare. Right now, that's provided by the hospitals. Most hospitals write off millions of dollars in care for people who can't afford the help they get.
I don't think people should get all their healthcare for free. Or even most. How people live affects their health. The option to pay for better care should always be available, and if we make it unavailable in this country, people will go elsewhere for it. See the canadian example. People who can afford to go to the US for their healthcare do. Profit is one of the incentives to better service and advances in treatments.
You don't have an inalienable right to be free from disease and injury. If you do drugs, and get hepatitis, I shouldn't have to pay for you to get a transplant. So if it's accepted that there are things that should be covered and some that shouldn't, then as a society, we need to determine where that threshhold is.
Right now in the US, people under a certain income are covered by medicaid. So most of the people who aren't covered by any insurance have chosen not to have insurance in order to have money elsewhere. That leaves illegal immigrants. Do we really want to set up a system where we treat anyone who comes into this country for any problems they might have, whether or not they're here legally?