Health care is a right.
Now, one could easily say - sure, so is a cold beer. Why should anyone deserve to see a doctor to get his blood checked. We don't have the right to food or water. We don't have the right to shelter.
But we actually do - when it threatens our health. Sure, we let bums rot under bridges. But if they are lucky enough to have someone notice them and call 911, then they are certainly guaranteed to have their condition stabilized at the nearest hospital. So yes, health care is a right.
Everyone is guaranteed health care access. The only difference is when and how it gets paid for. People with insurance get preventative treatment. People without it get expensive interventions. No one is turned away from the hospital and no one will likely ever be. People are, however, denied treatments that will either save their lives in the long run (and without costly yet futile emergency procedures), or denied preventative measures that prevent illnesses from ever developing.
How it is paid for is tricky. Depending on the hospital, city and state, uninsured patients are subject to different levels of accountability for payment. I'm not sure how much of the cost of the uninsured is absorbed by the system and how much is recouped from the patient. Certainly families often bear the brunt of expensive treatments the patient is unable to afford.
But should they be required to pay for their own care outright, especially if it is something that could have been prevented had they the means to access care earlier?
As a society, we can sort out who pays for what, via taxes, monthly payments, co-payments, etc. But everyone will get care.
My last thought on moral accountability: if our neighbor lies dying in his home, we are not responsible to save him. But if he lies dying on the sidewalk before us, we are required to do everything we can to save him.