One of the key requirements of any free market is free information.
Not quite. One of the key requirements of any free market is *perfect* information. There's no requirement that information be free in any way. Admittedly, it's generally improbable for information to be perfect and unfree--hence the discord of closed source software and security, among other things.
If you're familiar with "Medicaid oversampling" I'm guessing you're already affiliated with a health care provider. Are you currently pushing your provider to publish its prices? If not, why not?
No, I'm not familiary with "Medicaid oversampling" and Google doesn't really help there. As for pushing my provider to publish its prices, you make it sound like if I somehow got more information out of my health care provider I'd suddenly be able to get prices more in line with their own price estimates per insurer. Well, no, as another major point of the free market is a recognition that oligarchies and monopolies may be a natural consequence in a market place and as such they'll set their own prices which may fall out of an optimal* supply/demand point. As insurance is basically a large financial instrument where the more in the pool the lower per-user rates, it's rather obvious that insurance falls into the scope of natural monopoly/oligarchy. So beyond all the lack of free transition available to buyers, there's simply no means for natural healthy competition--even if cross-state insurance was allowed.
*Okay, that's a rub of the free market. If you're in a desert and there's one well with one owner and a million people about to die, yet no one has the asking price for a drop of water, then the "optimal" solution in a free market would be for everyone but the owner to die of dehydration. Hence, optimal in a free market and actually optimal for society or people in general may be very different things. And since we're having this discussion, I presume you are no more happy with the "optimal" solution that the free market tends towards in health insurance. No doubt, government interference may make the situation worse in many ways, but no government interference would have similar but different problems. Hence, either the whole system reasonable needs abolished or much better regulation needs established. Neither of which I see actually happening, especially as it's unclear how you can well regulate private health insurance when the wealth gap pushes people into free medicaid (admittedly, more often just the emergency room kind).