It is important that the law identifies residents for social security purpose ("belföldi) who enjoy additional benefits.
Residents for social security purposes ("belföldi) are:
Hungarian citizens with a registered place of residence in the territory of Hungary, immigrants, persons with settled status (meaning permanent residency permits) or persons recognised as refugees or beneficiaries of subsidiary protection (beneficiaries of international protection)
a person with the right of free movement and residence who has a registered place of residence (EU citizens)
stateless persons
Residents for social security purposes if they stop being insured or being entitled for TAJ Card (e.g. stopped working or their studies), as it is compulsory to be part of social security system, they are liable to pay health insurance contribution every month (if they have been have residency in Hungary for at least a year).
More information in HUngarian: https://www.neak.gov.hu/felso_menu/lakossagnak/ellatas_magyarorszagon/jogosultsag_az_ellatasra/az_egsszsegbiztositas_szolgaltatasaira_jogosultak/egeszsegugyi_szolgaltatasi_jarulekfizetes