A worthy goal, but the problem is (and always has been) how do we do this without bankrupting the country, making sure the care doesn't suck, and preventing abuse while keeping those who provide healthcare reasonably content.
I truthfully think one of the best ways to do this would be to have a government option (in addition to private insurance) to buy into essentially catastrophic insurance (say, 5k deductible) with premiums adjusted for income. Now unlike regular insurance, this deductible is applicable even if you pay cash for something. That's one of the biggest problems now with high deductible plans. They doesn't actually encourage price-shopping because it doesn't work like that. Everywhere will charge the same insurance allowable so its the same price everywhere. For example, I can get my patients a cash MRI for $500 at a free-standing imaging center in town compared to $2500 cash at the hospital systems. If you file insurance, its $2000 no matter where you go for it. So, patients are actually punished for trying to get a better price since the cash price won't apply to your deductible. Change that first off.
Then, set up an HSA-like plan. Tier the amount that the government puts in based on income so that everyone below the poverty line gets enough to cover almost but not all of the deductible (say 95% or something) while those at 2X the poverty line only get enough to cover say 75% and 3X 40%. Then, let everyone get their own health care.
Its the best way I can see to let the free market still have a role while not leaving out the poor.