I live in a country with free universal healthcare, for which i am incredibly grateful, especially since becoming a mother.
The problem with government funded healthcare is that it easily becomes a bottomless pit. You can always spend more. Science is constantly discovering new treatments and drugs, and every one of them is of vital importance to the people they can help. Hospitals can always be improved, more staff will always help.... the list goes on. What was considered a complete health system a generation ago is very basic compared to the system we have now. The costs grow yearly, and once committed to the principal of universally accessible healthcare they will always continue to grow.
I think the question that needs to be asked is "Is healthcare delivered to most of the population more effectively through taxation and government provision or through a privately operated user pays system?"
And i think that the answer varies from society to society. While government provided health care works relatively well in a small country like Australia (21 million people) it seems to become less efficient in larger countries such as Britain, with it's notoriously troubled NHS.
I guess what i am trying to say is that being committed to accessible health care for all doesn't necessarily mean a commitment to any particular ideology, be it free marketism, socialism, or anything in between.