If you or one of your family members is ever in the need of critical medical care and you would prefer to be treated at an Austrian hospital you can use an Austrian based ambulance service for pick-up at the Hungarian hospital and transport to Vienna. This organization has Ambulances available that can transport both adult and child/infant intensive care patients.
Hungarian ambulances will only get you as far as the border.
Please note that it is your responsibility to make sure if this service is covered by your insurance provider.
Website: Gruenes Kreuz in German only
24 hour hotline: +43 1 14849 (English speaking operators)