If the thing we all have in common is that we pay for each others' health care, then all our health care is done. We don't have to worry about it. We sort-of already do this, by the way, with insurance, but because the insurance is profit-motivated, it doesn't accept as much risk and is exclusionary.
So we already have a collectivist health care system mapped into capitalism, with some middle-men ending up really rich, at the expense of those whose conditions are risky to insure.
If my parents can't afford insurance, I end up being tied to them in a way that demands a sacrifice of my creativity. And the only reason this occurs is because some middle-men get to set the terms of the insurance policy, so that they can make money.
Taking the profit motive out, we end up with insurance for whoever needs it. Suddenly, my parents aren't uninsurable any more, and I am free to support them to whatever degree I choose, allowing me to maximize my individual economic liberty.
The middle men don't like this fact, and they'd really like to keep their stranglehold on my creativity. But should we let them?