With regards to the Underworld Squad (they need an actual name ik, I'm working on it) I basically made a list of the Gods I'm counting as Underworld deitites/Gods.
Hades/Pluto, obviously. He's the king of the Underworld, and Nico would head this squad so yano.
Persephone/Proserpina. Married to Hades, spends half her time way down Hadestown (sorry I had to) and is the queen of the place, so I'd count one of her kids as an Underworld demigod.
Thanatos/Mors. He's literally the God of death, his kids are going to be seen as nothing but bad omens, and then be an outcast so they'd love a squad where that isn't the case.
Melinoe/Meliana. She's the Goddess of nightmares/insanity and a daughter of Hades and Persephone, so the element of the Underworld is there. Nightmares are constantly plaguing demigods, so a child of hers would be outcasted (is that a word?), and insanity has the same sort of foreboding as death and stuff.
Plutus/Ploutus. God of wealth, and a son of Hades and Persephone. He has the Underworld part covered, son of Hades so that's that, and there's also the wealth part. We know from Hazel it's still seen as a foreboding Underworldy thing, so the outcast box is definitely checked.
Geras/Senectus. God of old age/aging in general. Demigods tend not to live a long lifespan, so a child of the God that puts them closer to that supposed fate is going to creep them out. He's a son of Nyx/Nox, who resides in Tartarus, which is a part of the Underworld, so I'll count it.
Speaking of, Nyx/Nox. Honestly a child of a goddess as powerful as Nyx would instantly be seen as a bad omen. Like, a primordial deity who resides in Tartarus, a place that scares gods, having a child at Camp? That's like when Nico, Bianca and Hazel all showed up for the first time but a lot worse.
Akhlys/Achlys. Again, resides in Tartarus so that's instantly going to be seen in a negative light. Also, a child of the frickin Goddess of misery/poison? Likely to be stereotyped as a killjoy/Debbie downer and not given chances by other campers so they find Solace (pun intended) in a crew of kids like them.