HI Pusinka,
as you're an EU citizen, you're entitled to Slovak healthcare. My accountant advised me to also have a private health insurance, as the quality of healthcare is much lower compared to central Europe. An insurance with repatriation to the UK would be advisable.
By the way, my accountant told me also that private health insurance doesn't exists in Slovakia, so you'll have to find one abroad.
Regards,
Gregory
