Are you sure about the name "Emperor's Prodigy"? Sounds a bit... harmless for a ferocious hellenic-oriented Space Marine chapter. I don't think there is a ancient Greek word for prodigy.
Emperor would be Basileus (ΒΑΣΙΛΕΥΣ), meaning "king", or maybe better Autokrator (AYTOKPATΩP). The genitive would be Autokratoros iirc; so the name would be "[prodigy] Autokratoros". You could of course do a Greekish High Gothic version of it, calling them "Prodigis Autokratoris". That's the fun with WH40K's Imperial Gothic: It's absolute bollocks and to someone who knows about Latin it sounds so too but to everyone else it sounds totally "WOW". :D

"Squad" being the smallest tactical unit as I understood it, I would label "taxeis" (basically "(soldiers in battle) formation"). You could name your assault/devastator/whatnot squads after hellenistic elite heavy infantry, like Pheraspidai, Hypaspistai, Argyraspidai etc. Would be cool, I think.

Tank... very easy: Kataphraktos. :D