Hungarian healthcare is publicly funded, but there are private institutes as well.
Foreign citizens may be eligible for free public healthcare services only if they come from countries that have a special agreement with Hungary. But even in such cases, public health care services may only be provided up to the necessary extent, in case of acute illness or emergency.
Private healthcare is fast, professional and of high quality. On the other hand, it is not for free. You should always have accident and health insurance before starting your trip. When choosing an insurance plan, always check the services it will make you eligible for.