The more insidious thing about universal healthcare is that it really screws over most minorities. Minorities tend to have diseases that are not as well researched as those found in ethnic majorities (or certain minorities which are disproportionately represented in the medical research community). Moreover, drug testing -for safety and efficacy- in minorities is very scant, and it is sometimes the case that adverse reactions to drugs occur in minorities that do not occur in the ethnic or racial majorities. Of course to find that out you need statistically relevant sample sizes, and the way drug safety testing is done is by recruiting on college campuses - you can see the potential problem here - and even still some minorities that are overrepresented on colleges are, for whatever reason, underrepresented in testing pools. So as a minority, one might be paying into a system that disproportionately treats diseases that one would not be getting, or with different pathologies, and for which the treatments may be completely ineffective or even dangerous due to lack of study.
Then there is this problem. What if those who decide health coverage decide that heterozygous genetic condition X is covered (and it's expensive). Two parents who are carriers now can decide, hey we can afford to have kids - after all it's their right. So now everyone is saddled with the burden of their selfish choice. Or, we can restrict their liberty by prohibiting all such pairs of parents from reproducing. Whereas if they are responsible for the cost of their children's care, then the simple answer is they might think twice about having children and if they are foolhardy enough to, there is a 1 in 4 chance they get dinged with the cost of coverage.
Ultimately, any governmental (i.e. non-free-market) health insurance plan must either explicitly discriminate between individuals or, if universal, implicitly discriminate between individuals. The perverted consequence is, ultimately, going to be a transfer of wealth from the poor (via inflation) to those who have the right genetic predisposition to conditions which are decided to be covered by the powers that be.
Keep in mind that 'free market' != 'for-profit'. You can have nonprofit insurance companies, and healthcare charities, in a 'free market' system. What free market means is that the consumer has a choice of where to go for their services. If you are saying that you don't like 'free market healthcare' you are saying that in the aggregate, the population is too stupid to make responsible, cost-effective decisions about their own health. If that is the case, who or what group of people could possibly be so enlightened and all-knowing to correctly orchestrate all those choices from the full society on down to the individual level? And how might we shield society from the hubris of our godlike health chancellor/council?