Square handled the PS1/GBA ports/remakes of games but after that they had others handle it. To be specific, it looks like Matrix Software handled a lot of the FF ports/remakes (notably for iOS, Android, and the DS).
They did a great job on the DS remakes of 3 and 4 though (I highly suggest both, if you have not played them....4 moreso than 3), so if they handled a compilation I wouldn't be worried about the quality. Their version of FF4 is, imo, easily the best version available (the Steam version is technically the definitive version of it, imo, over the DS/Mobile versions but they are the same game really). My only gripe with them is the graphics style they chose for the iOS and Android versions of some titles (such as FF5 and FF6). I feel that the "HD" versions made the characters less detailed, honestly. They made them look a lot more cartoony and more...flat. The original pixel-art just had a lot more charm to it. But that's just my opinion, others seem to like it *shrug*.
So yeah it would easily be a viable option to outsource it to Matrix like they have been, though I just don't see it happening anytime soon. One can dream, though...one can dream.
(Note: If you get the Steam version of FF4 you may want to look into a couple mods, though. They did a LITERAL port of the mobile which means the FPS is the same as it would be on mobile, hitting 15FPS in battle on PURPOSE that for some reason they carried over to the Steam port. You can get a mod to fix that and unlock the framerate, however)