But what I'm wondering now is, if you took an Ace: Gunner specialization and then bought a specialization in, say, Technician: Outlaw Tech, would the same character have to have both Obligation AND Duty? Or can you decide which, or both? And is there anything from saying you can't take Duty instead of Obligation for a character career from the EotE book, or vice versa? Can I have a Colonist: Doctor who has no Obligation, but a Duty instead?
I'd say no.
Just because you bought an EotE specialization doesn't mean your AoR PC now suddenly has Obligation, and your EotE wouldn't be required to sign-up with the Alliance just to take an AoR specialization.
As for using EotE careers in an AoR game, substituting Obligation for Duty, I don't see why not.