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.
Things 
This reminds me of a Universal Justice question on okcupid: “If a police officer is found guilty of a crime, do you believe the punishment should be more or less severe than for the average citizen?”
By assuming a queue of 100 people, you assume that hospitals are shitty to begin with, though. Also: a doctor who gets treated does not automatically become ready to treat the other 99 patients in the queue (except if he his problem was a slight headache perhaps).
“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.”
I can imagine: Nobody in that line would get treated ever, because everyone has some important function in society, and they would argue to the end of days. So maybe we should establish a rule that establishes fairly who gets treated first…. how about: Whoever came in first should be treated first?
Sure, you can do this, but this is the system that minimizes arguments between sick patients about who should get treated first, not the one that maximizes the overall health of the society. The point of this post was to demonstrate that maximum health and maximum fairness cannot be achieved simultaneously. Or in other words, if you want to maximize the total health of the society, a first-come first-served system is not what you should implement.
But since arguing in this scenario is what is keeping the patients from getting healed – wouldn’t you also maximize the overall health by minimizing these arguments? If they have the choice between arguing forever and getting healed one by one, shouldn’t they choose the second option?