The answer is “it depends”…on your particular circumstances and the country you are moving to.
With so much of Europe unified, it is no wonder that expats expect that there is a European system of healthcare. But there is not. Each EU country has a different social security system, including healthcare. Most countries offer universal healthcare through either a socialised medicine or multi-payer system. The standards do vary between countries, but overall the standard is good.
Many expats despite moving to countries with exceptional public healthcare systems, still choose to take out healthcare insurance to ensure their needs are immediately addressed.
I’m sure the folks at Overseas Property Forum can help. They helped me source my cover for Greece.