The trouble is, if health care is in the hands of private interests, how far they can be trusted with a person's well being. As with any business, private health care providers are seeking profit first, rather than health being the main priority. This entails a whole host of problems, i.e. medication being proscribed that attacks merely the symptoms of a disease rather than the root causes, the poorer in society unable to afford decent health, etc.
At least in government hands, one has at least some semblance of accountability, whereas private businesses are lucky if they get convicted for corporate manslaughter.