Thanks for the question.
Get your dog chipped with an ISO-compliant chip. If any of your dogs have more than one chip, the number and date of implant must be made clear on the EU Health Certificate.
If you are moving from a rabies-free or rabies-controlled country, your dog will need evidence of a current rabies vaccination. This must have been done after the microchip was implanted. The pet will need to wait 21 days before travelling if the aforementioned vaccination was its first one.
Importantly, if you are moving from a high-rabies country, your pet must be vaccinated after being microchipped. It is important to do it in that order. A titer test will then be administered to your pet 30 days after the vaccination, with results being cleared at approved laboratories. Then, if the results are within acceptable limits, your pet can enter the country three months after the blood was drawn.
Something to note is that, like any country, Hungary has a list of banned breeds and you should check that out too.
There are a few general articles on this topic which you can find on the web. I think, in order to be safe, you should contact the Hungarian embassy in your region and ask for a checklist of requirements.
Good luck to you and your dogs!