There is no evidence that universal healthcare in America would be cheaper. In fact given the data we have the opposite looks true- the fact that we spend so much on Medicare and it only covers a small part of the population.
The real problem is the American demand for instant gratification and consumption of any product including healthcare
Until we can learn to accept "no grandma can't have that procedure that will only give her an extra week of life at a cost of a half million dollars" or "no you can't see YOUR doctor until the people in line before you see one so here is a nurse instead" we are screwed when it comes to the cost of health care.
The greatest difference between our cost of care and other civilized nations is they control those costs from above. The swiss for example use private insurers (w/ attendant overhead) but cantons set the prices, and they still pay substantially less in aggregate despite one of the most expensive places to live.