Europe is big, made of many different countries, with very different policies on public health care and very different standards for that public health care. For example, in Romania (part of the EU!), the big cities usually have good hospitals, but if you're in villages or smaller cities, good luck! Also, to be able to benefit from free public health care, there's a list of conditions you have to meet, it's not available by default to anyone. The requirements are quite reasonable (be employed, be on maternity leave, be in school, etc.), But there are people who are not covered.