US/Polish Citizen Retiring in Europe.

I have a dual citizenship in USA and Poland. 
Born in Poland but worked in the USA for the last 30 years.
I want to retire and rent a long term place somewhere in Europe, not necessarily in Poland, but I'm little confused on the health insurance and EU residence rules.

I know I can buy into Polish Health Service System with about $3k in pre-payment fee (since I never worked in Poland and did not pay into the system) and then yearly ongoing fee.
This would then (most likely) be transferable to other EU countries health systems.
For this to take place - I would need to be residing in Poland to begin with.

Question:  Since I am EU citizen - can I do the same in other countries?  Greece, Spain, Italy, France?
Can I establish residence directly in one of those countries and buy into their Health Care System?

Would there be any other legal issues to consider?

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