I would keep the private sector legal and not allow people making over a certain amount access to the government run health system(without any tax break).
This part of your statement takes you beyond left wing into the far loony left. You are not only forcing people to pay for other's bills, but on top of that you are prohibiting certain people from participating in a system that they are being forced to pay for.
Why don't you just walk into their houses, steal their money, and be done with it?
They get to pay for thier own healthcare, which is what they want. In exchange for taking advantage of the poor by getting better healthcare they have to pay the penalty.