Yes, in some places. In the US the entrepreneurship rate measuring the % of the population involved in that kind of activity hovers around 14%-ish. There are European countries like Estonia and others in the north and baltic regions that beat that rate while still providing high quality universal healthcare. As a rule, though, it seems like western European countries are significantly less entrepreneurial.

I think the entrepreneurial rate has a lot more to do with cultural attitudes around risk and individual success. Some countries like Italy, for example, have rates in the very low single digits. I'm not sure you can overall make a case for universal healthcare increasing entrepreneurial enterprise. Whether it might increase it specifically in the US might be a different question, though.