The EU/EEA has a coordination system for social security (which includes national health insurance). One of the fundamental rules of this system is that you are always subject to exactly one national social/health insurance system. Which country you are insured in is determined by a set of rules or (in complicated cases) by a decision of the relevant national authority. You do not get to choose where to pay your health insurance.
As a few examples, you will be insured in the country where:
- You work (if you work as an employee or a self-employed person in only one country)
- You live (if you work in multiple states simultaneously)
- You worked last (if you're unemployed and e.g. receiving unemployment benefit after being employed somewhere)
- You live (if you're unemployed and not receiving any benefits or you have never worked anywhere in the EU/EEA)
The country where you are insured will also issue you an EHIC card. This card entitles you to urgent healthcare in any other EU/EEA state. It does not give you full health coverage (including non-urgent care) everywhere. However, in some cases, you can be fully covered in multiple countries at the same time, for example if you're a cross-border worker (returning home at least weekly).
Depending on your exact circumstances and your current status, you may first get a Hungarian EHIC card which you would then have to return when you move to France, where you will get a French EHIC. Or if you're going to move really soon, you may get to skip the Hungarian part completely.