I want to run the argument the other way: the right to health care exists, but is just like the right to keep and bear arms, it entails neither the right nor duty of the state to pick my neighbor’s pocket to provide me with the means of exercising my rights. Obamacare is an infringement of the natural right to health care, just as requiring enrollment in a government approved arms-bearing body (army, police, etc.) to bear arms is an infringement of the natural right.
No real “right” exists that requires an involuntary burden on another person.