Hey! Sorry it took a couple days. But potentially some good news!
So I hadn't used my MA database in a while, and as I was poking around to try to answer your question, I realized it needed a software update. Sadly, the update failed. So I just went ahead and downloaded a whole new ZIP of the latest version (1.5) and pointed it to the workspace folder for my old MA install. Everything worked like a charm. More importantly, I noticed a couple things:
First, it was asking to update its card database, and it included all of the recent editions, including WAR and M20. So you might be in luck!
Second, if you're still interested in using Deckbox, this new version of MA has an Export option (I don't know in which version it was added, but my old version didn't have that option), and you can set the "Export type" to "DeckBox Inventory CSV". That'll get you most of the way there! But when you attempt to import it into DB, you'll probably receive a long list of errors due to mismatching edition names and special characters in card names (like the Æ in Aether). You can either go through the errors and fix them yourself, or if you'd like, I wouldn't mind taking a look at the MagicAssistant-exported CSV and see if I can get it 100% compatible with Deckbox for you.
If for some reason you're still not able to get a new version running, so that neither of the above are an option for you, then go to your Workspace directory (if you've forgotten it, you can view it in Preferences > Magic Prefs > File Locations). In your Workspace directory, open magiccards > Collections, and attach your collection file(s). They're in XML format. I can't promise how easily I can convert those, though, so trying again to get a new version of the program will be your best bet.