First of all sorry for my poor english :)
If you want to establish a sole proprietorship in Greece one of the required actions is to register with the appropriate social security department which in this case (for self employed people) is called OAEE. After that and as long as you are consistent with your insurance contributions you have all the benefits for social health care. The establishment of a sole proprietorship in Greece is a little bit COMPLICATED -be patient! :) - for bureaucratic reasons. You will need the help of an accountant.
Look here http://www.atskakanis.com/wp/?p=336 for some more infos.
VERY IMPORTANT!
The ragistration with OAEE and the payment of the insurance contributions is mandatory. Of course the same time you can be registered and with a private insurance company if you want.
Dont forget to ask if you are entitled for any subsidies. There are some european finance programs for self employed people who establish a NEW company or shop for which and under some conditions you may be eligible.
Good luck from Greece!