Good one.
FYI, there already is a solid government/free market combination in effect that is popular with its participants: The Medicare / Medicare Advantage / Medicare Supplement system, that provides seniors with a solid foundation and FREE MARKET choices, competition and innovation. It could easily be tweaked to cover everyone. There are several moving parts that can be revised to keep costs down.
For example, here's an idea of mine: The Medicare portion can be smaller when people are younger and healthier, and they'd then have options with FREE MARKET Medicare Advantage plans or FREE MARKET Medicare Supplements. Those would cost less than they do now because, again, the person is younger and healthier. Then, as we age and become more expensive to cover, the government portion can increase to what it is now, and the participant can still get a supplemental policy.
Right now, our health care "system" (ha) includes seven (7) different service delivery/payment systems, NONE of which communicate seamlessly with the others : Individual, Group, Medicare, Medicaid, VA, Worker's Comp and indigent. That is fucking INSANE.
Everyone would have their own portable, personalized plan. Imagine the increased efficiencies of not having seven (7) different internal systems. Imagine the cost monkey it would take off the backs of American employers. It's just sitting there, ready to go (with some doable tweaks), but our "leaders" (ha) are no longer allowed to communicate, collaborate, or compromise in any way.