I think some level of basic emergency care, and care of chronic conditions (to some set degree) would be a good thing to provide everyone. I think it would have to come with some strong limits on what is covered, to the point that the government may not cover some life-saving procedures if they're too expensive, or don’t provide enough return in terms of quality of life. Those procedures would have to come out of the citizens (or his insurance companies) pocket to avoid bankrupting the state.
From what I've heard (very third hand) about other countries systems they tend to offer every available treatment, but also tend to have supply shortfalls. Are those because there's no budget for the procedure, or the government is just not good enough at predicting demand?
If budget is the problem maybe the solution is to have the government cover fewer procedures/pills, so that you can more effectively guarantee access to what you do cover. Leave the more exotic, lower cost/benefit procedures to private industry and cover the comodity type healthcare inside government.
Going a bit more general..
The problem with healthcare is that when you’re sick and/or dying you can’t really put a price on a cure. You'll pay anything and everything you have, because if you don’t your quality of life is seriously degraded, or worst case you die. As technology gets better and better this problem gets worse because there are newer and more elaborate (and thus more expensive) ways to maintain your good health. The problem gets worse with insurance companies, since it’s not your money you spend when you go in for that "must do" procedure.
If you decide to regulate the entire system and provide universal care, you'd have to find a way to set a maximum price that is worth "investing" in an ill person. It sounds cruel, but if the government across the board starts spending its resources pulling every last string to keep every last person alive and comfortable, the bill will be more than taxpayers can bear. Take it further to "elective" procedures, and you have to ask questions like how much Viagra does the government pay for per covered citizen?