With Public Healthcare there is no middle-ground. Either everyone in the country has it, or noone.
Let's compare it to fire insurance for your house. In that case not everyone needs that, only the people who want it. Should your house burn down and you haven't got fire insurance, tough luck; your house is gone and I hope you have enough money but you need to take care of it.
But the deal with PH is a whole lot different. When you need medical threatment or else you'd die, we don't say tough luck. This is because we (most of us) are civilised and won't let other die when they could be treated. One problem however, that surgery you just had to save your life cost you 100.000 dollar. You don't have this--> means the goverment has to pay for it.
Let's say everyone does have PH insurance, your surgery still costs 100.000 dollar, but now this will be divided among EVERYONE. Sure, some people won't need health insurance, they will never get seriously ill. You will however not know this (else please tell me the numbers for the lottery).
I think there are 2 options:
1) Health Insurance is not an option but an obligation
2) It is an option, in the case someone doesn't have it but needs treatment you throw him on the streets, he doesn't have the money after all.
We don't want option 2.
I myself live in the Netherlands and we have a universal healthcare system here, which granted doesn't always works perfectly, but comparing it to what the USA has it's divine.