In Europe there are countries where you can not be a part of the public system in some cases, such as changing your permanent residence outside the country.

Well, ok, I see what you're getting at - but unless you moved to outside of EU, you are still fully covered by the country where your new permanent residence is. If you moved out of Czech Republic to Spain for example, just get an EHIC card from Spain and you can get treatment wherever you go in EU, including Czech Republic.

Yes, but I was talking about the case when you're not insured. Btw the treatment you're entitled to in other EU countries (outside the issuer country of your insurance card) is limited (but of course childbirth is covered) and you need travel insurance (it's cheap in EU, around 2 euro per day), especially if you're doing sports as any injury done during sports is not covered at all outside your country.