It doesn't seem obvious at first. In fact, it seems contradictory. How could it be possible that a free market system could co-exist, and even benefit by, a government run, single-payer, health care system?
Let's first of all establish what "free market" means. I've received a recent and inspiring education from Rational Reasons and a bunch of other sources on the subject of "free markets" and "libertarianism".
If we were living in a free market system (which we aren't) it would mean that consumers are free to choose the best products. Natural economic supply and demand would work together with the consumer's judgment of quality to select the supplier or manufacturer who is doing the best job. If we impede the flow of the free market too much - say by creating incentives for the existence of large companies or by subsidizing certain raw materials - we will damage the ability of the free market to adjust itself to maximum efficiency.
On paper, you have to admit that this sounds pretty good.
As an example of where the free market would fail, if every soap I can buy changes my skin complexion such that I have to go through a four week period of skin rashes and boils when I change soaps, we can't really pretend that there's a free market in soap.
My argument is that, in a free market, people have to have the same freedom to move about from job to job. If my employer can keep me attached to him unfairly – say by blacklisting me, beating me up or burning down my house – then we don't really have a free market. It would allow terrible employers to continue to employ people that the employers don't deserve to have employed.
This is how I see private payer health care, as an unfair tether.
The fact that health care is provided by an employer allows an employer to unfairly attach me to his company. He can pay me less than a proper market rate - therefore damaging the power of the free market - simply by the fact that I could find myself dead if my next employer's health insurance declares my disease to be "pre-existing". Even worse, the situation creates a powerful bias against small companies and self-employment because private health insurance becomes prohibitively expense for individuals and small groups of people. This creates incentives for the existence of larger companies that otherwise might have no business existing in a free market.
The only possible way to keep the free market going is to remove the private money from health insurance. Health insurance should be paid by a government. Even if you wanted to allow private clinics to provide services in competition with the government, you have to keep private money as far away as possible from the paying side of the operation.
This is the only way to guarantee that labour is sufficiently lubricated to make this whole dream of the benefit of the "free market" come true for the vast majority of the people.
Guarantee people health care and you guarantee them freedom.
The mantra from the libertarians is that it is the freedom to choose that will make the economy work properly. In order to really have the freedom to choose, the people must know that, regardless of their choice, they will be secure of body and health. Consequently, we must sacrifice a small, irrelevant part of "choice" (i.e. who pays for the health care) in order to open up a whole world of choice in something orders of magnitude more important - where you want to work.