I genuinely hope they don't. I do think they'd make for a lovely pairing (And given the collaboration on FF exclusives it does seem like a natural progression), but I honestly don't want to see Final Fantasy and Dragon quest and other Square-Enix games like Bravely Default and Octopath Traveler NOT on Switch.
I do like sony and like how they generally treat their studios but I just don't see them publishing on Nintendo anymore if they acquire Square-Enix. And I think that'd be bad for the industry.
For Dragon Quest, at least, it wouldn't be entirely Sony's decision, from what I can tell.
I'd imagine DQ would remain on Nintendo devices, but just no exclusive releases. So, PS, Nin, and PC for all DQ games.