We moved to France when my husband retired from the armed forces. Since then things have changed and it is very much more difficult for people to move to many (all?) parts of the EU unless they are working or have a state retirement pension.
In 2004, the EU issued a directive that covers your rights and obligations when moving to other countries in the EU.
Most countries seem to have adopted them. I don't know what the actual situation is in Greece at the moment, so it is as well to find some Greece specific info but I would check that anything you find is bang up to date. I am very aware that national regulations can change overnight*
If you are a pensioner you may live in any EU country if you have:
a)comprehensive health insurance cover there
B) sufficient income (from any source) to live without needing income support
If you move to an EU country and are in receipt of a UK state pension (occupational including from the armed forces, police, teaching etc are not state pensions). After obtaining the correct documentation, form S1, you can join that countries scheme and the UK will pay a contribution for you each year. What the local health service provides varies from country to country and there may be some costs
If you retire early with an occupational pension then to move to another EU country, you have to be able to show the means to support yourself and also that you have comprehensive health insurance.
if you have been paying NI, you can obtain a form S1 from the UK (dept work and pensions) that will allow you to join the health care scheme BUT this only lasts for a maximum 2.5 years. This is dependent upon the time of year you move and your contribution record.
After that you have to have private insurance and sorry,I don't think it is possible to get such insurance with T1.
If you manage to get affordable insurance then after 5 years legal residence then you have the same right to join the offcially scheme as a citizen of that country. That may mean, in fact almost certainly does mean paying a contribution.
You would be able to join the countries health care system by working there, or by becoming self employed and setting up a business and paying contributions. You would need to get country specific advice on that.
*the French did change their regulations overnight and for a while they were going to implement things retrospectively, this caused me and many others a period of anxiety. After a great deal of lobbying by a group of brilliant group of activists
they changed their minds.
However the EU regs apply to new entrants to this country.
EU directive 2004/38EC - Amended
http://eur-lex.europa.eu/LexUriServ/sit ... 350048.pdf
edit there were different rules for people receiving incapacity benefit , they used to be able to get cover through a form issued by the UK but things have changed a lot in the UK recently and I don't know how things stand for people in this situation.