Today hospitals need patients with health insurance to subsidize everyone else. Private health insurance pays about 150% of the hospital's actual cost to hospitals in reimbursement, whereas Medicaid and Medicare only pay 50-90% depending on the state. Of course patients without health insurance often end up treated for free: they pay 0% of cost. Hospitals need to prioritize patients with health insurance to ensure they have funds to pay for those who don't have it.
The sad reality of the situation is that hospitals are not charities. They cannot afford to give their services away. When people without insurance are treated, the cost of that treatment is simply passed on, it doesn't disappear. This means either higher costs for those who do pay or higher taxes for everybody. Eventually this also means the hospital is forced to close, meaning that nobody at all gets service. We need to focus on reducing the cost of health care, so it's more affordable for everybody.
A hospital can't afford to take on people that can't pay their way (if they make it clear they can pay with cash, that's ok). The only healthcare that is a right is preventative. Treat your body right and you won't need a hospital. You can't offer anything for free that is a service, especially one so time intensive. Doctors and such professionals have some of the largest overhead, and highest insurance costs (malpractice) of any business. They go to school for most of their young years (probably be about 30-35 when they make it out). Any acknowledgement they get for this hard work is well deserved. Moreover offering "free" healthcare just means someone else pays for it.
No, I do not agree that private hospitals should be able to place priority in providing their services only to patients who have health insurance. If someone comes through the door and is bleeding to death or dying of a heart attack, that person is in urgent need of care, whether or not they have money. Health care should be a human right, not a privilege.