Who have their permanent residence in a Member State of the European Economic Area (EEA) are entitled to use healthcare services in Hungary through thr European Health Insurance Card (EHIC). More information on the EHIC can be found here:
The biggest private health insurance providers in Hungary are Medicover and Cigna: http://www.medicover.hu/medicover-health-insurance/?lang=en
In respect of European Economic Area citizens social security EC Regulation 883/2004 is ruling. This defines the social security status of the international employees in the EU countries. This regulation changes the earlier system, based on ’E’ documents (certificate of the social security coverage in a country). The new system is valid for medical care, sick-pay and pensions and defines an internationally mobile worker’s social security contribution position within the EU. There are new requirements for remaining insured in the home country for social security purposes, notably where an individual works simultaneously in two or more member states. Before travelling please get information from your local social security authority about your coverage status while you are working in another country. Also ask your Hungarian employer before about the possibility to get a social security number/card (TAJ number/card) in Hungary. If you have this number, you are entitled for full medical care and sick-pay in Hungary and will not need a supplementary insurance (over the European Health Insurance Card). The regulation is available here.
In case of other, non-EEA country citizens working in EU also the EC Regulation 883/2004 is ruling. If you are arriving from countries with which Hungary has a bilaterial social security agreement - Serbia, Macedonia, Bosnia-Hercegovina, Montenegro, Kosovo – you do not need to pay social security contribution in Hungary, but you can get a Hungarian health insurance number/card and therefore are entitled for full medical care and sick-pay. In case of all other non EEA country citizens social security contribution must be paid in Hungary in order to get the Hungarian health insurance number/card
You can find detailed information on the social security rights avaliable to you in Hungary, along with other EU member states, at the portal of the European Union.