At a minimum, they'd get some core spells from their class quests, which are themed around going and learning them from monsters. As mentioned above, they could stop there, making BLU like any other class with a slightly different flavor.

Ironically, BLU would work better in Guild Wars, where the game is made for classes to swap abilities in and out. If something like that could be copied and adapted to FFXIV, it could work well, but it would be a ton more effort than any of the other dozens of classes they could add, so we're not likely to ever see it unless it's a simplified version.

I'd say the best we could possibly hope for (and even then it's a bit much) would be learned BLU skills going into the 'additional' category and being equipped to cross class slots in lieu of taking skills from other classes. The abilities would likely be monster themed versions of existing cross class skills, with maybe a few exceptions.