The following argument shows that the most extreme form of universal healthcare, i.e., the one where every individual is given exactly the same kind of healthcare, is impossible, assuming you also want to provide the best healthcare. Universal healthcare can obviously be achieved if you don’t care about its quality. For example, no healthcare at all is horrible, but it’s at least equal for everyone.
So, for contradiction, suppose it’s possible. So there is a hospital that treats its patients on a strictly first-come first-served basis. No matter how much money you have or how important you are, if you come in last, you will be treated last. It is easy to see that this hospital provides horrible healthcare. Why? Imagine that there’s a queue of hundred patients and then a sick doctor comes in. If they treated the doctor first, they would have one extra doctor to now treat the rest of the patients. So by not treating the doctor first, they are providing inferior healthcare to the people already in line.
This argument can be gradually extended from the doctor to, say, a doctor’s secretary. If a doctor’s secretary is sick, and the doctor needs the secretary to take care of some tasks before he can start treating patients, it seems like a good idea to treat the secretary before the other patients. How about the secretary’s secretary? You can imagine where this is going.
So essentially, even if your only aim is to maximize society’s health, some people must get better healthcare than others. Equal healthcare and optimal healthcare cannot exist simultaneously.