That stuff is in PokemonBoxManager, a strange and mysterious script section which I think imports and exports information to ROMs of the official games (including information about currently available Mystery Gifts). It'll all be out of date by now, and I don't know how to work it in any case. I may well scrap it, because I'm sure no one's ever used that feature (and it'd be cheating, not to mention it promotes the using of ROMs which legally we shouldn't do).
No, there's no Mystery Gift function in Essentials.