As is so often the case, the devil here is in the details. In other words, I would be willing to pay a good deal more in taxes for a universal health care system that actually provided good health care to everyone and which made moves to rein in the rising cost of health care. However, I would not be willing to pay much extra at all for a program that would not reduce costs and/or would not really provide care to everyone.
I do think that it would be a good thing if everyone in the US had health care. I am willing to have the government pay for it. I would be willing to pay as much as (I think) 20% more in taxes than I do now to fund this. However, it would have to be a program that was well-crafted. I think that ideally it would be a program that set up a free market system in which health insurance companies could compete to attract customers. In other words, I do not think that it would be good to create a fee-for-service program like Medicare, only on a larger scale.
So, I would be willing to pay a lot more tax, but only if the program were set up in a way that would be beneficial to the country.