My $0.02 on the matter (which I'm sure will make for an interesting discussion) is that I agree, everyone already has access to health care at some level, whether you have insurance or not.
Given that those without insurance that can't pay for it still get care at some level, that gets paid for by someone (say, the hospital they get service at.) The hospital has to get the money from somewhere, and that somewhere is from people they service who do pay their bills, either directly or through insurance.
Now, if that somewhere didn't have to cover the extra costs of the someones who don't pay their bills, those that do pay their bills would have smaller bills (again, ignore the for-profit greedy pigs at the hospitals, assume the system is working properly.
Thus, in my mind, I'm already subsidizing those without insurance and/or that can't/don't pay their bills through my higher bills. And these usually come down as more expensive triage/catastrophic type environments because people go to the emergency room for a cold, etc.
So, in my mind, in order to keep more of my own money, I would rather subsidize people that can't afford to insurance or pay their health care bills with proactive health insurance that helps keep them healthier, in less catastrophic environments, which (again, if the system was working properly) should reduce my costs as well because I'm subsidizing care in a cheaper preventative environment instead of subsidizing care in a more expensive catastrophic environment.
Again, my $0.02, and if the system worked properly. yeah, yeah, yeah, health care utopia, not gonna happen...