It is difficult to advise on the best choice of technology without knowing what you are persisting and why.
If the state is simply a snapshot of your game state (i.e. a save file) or a "best scores" table, then you don't need a database. Serializing using JSON, XML or ... Java Object serialization is sufficient.
If the state needs to be read or updated incrementally or shared with other applications ... or users on other machines ... then a database is more appropriate.
Serialization mechanisms are problematic if the requirements include incremental changes, etcetera. You end up building a database-like layer over the top of the serialization.
As to whether you should stick with Java serialization ... or switch to JSON or XML or something like that:
Object serialization is simple, but it can be fragile if you change the classes that you are serializing. This fragility can be mitigated, but it is messy and you lose the simplicity. (You need to write custom readObject and writeObject methods that know how to read "old versions" of the serialized objects.)
JSON and XML are a bit more complicated, but still relatively simple if you use an object binding mechanism.
It is worth noting that changes to the persisted object classes (or the database schemas) are potentially problematic no matter what you do. There is no easy universal solution to this problem.
UPDATE
Given the additional information that you provided in your first comment (below), it seems like you don't need a database in the game itself. All you need is something that can read and analyse the session state save files that your beta testers provide for you. Indeed, it doesn't even seem like the actual app needs to be able read the files. (But that's unclear, because you've not said what the real purpose of these files is ... or at least, not what the entire purpose is.)
It is also worth noting that you are probably saving the wrong information if your aim is to tune the sets of questions. What you really need to do is record the length of time and whether the user got the right or wrong answer and the time ... for each individual question. And you probably need to know what the actual answer given was ... so that you can spot cases where the user's answer was actually right and you "marked" it as wrong ... or vice versa.
"What turns me away from that idea is the question of how am I going to write their database into mine (in the future)?"
Exactly. If you hadn't prematurely "analysed" the data, you wouldn't have this problem.
But ignoring that, it seems like that a simple state saving mechanism is sufficient to meet your (still hypothetical / inferred) requirement of keeping a personal score board for the end user. Your "tuning" stuff would be better implemented using a custom log file. I cannot see any value in incorporating a database as part of the app itself.
