Why does the question assume that universal health care requires single payer?
I'm not sure that it does. You can be in favor of universal health care but not single-payer systems. In that case you'd vote No in this poll.
We had a thread a while back that compared four basic medical services models. The conclusion was that the United States was unusual because it maintains so many separate systems for separate classes of people, utilizing aspects of all four of those models. Apparently all the other countries have settled on one model for everybody, and any of them seem to be simpler and less expensive than the US system.
I voted No as well, for what it's worth.