The only case that would maybe trigger a payment would be if you were a visitor to EU and had a baby - but most likely no hospital would bill you as hospitals are just not equipped for billing, they don't have the staff or the equipment to do such weird things as issue bills for treatment to people. I know at least several non-EU people who had treatment in EU and no one ever asked them for any insurance info to bill them, unless it's something major and very very expensive there's just no point, it would most likely cost the hospital more to figure out how much and how to bill you in the first place.