Thanks for that background. I watched
Hi, Mom!
without having seen
Greetings
, so I assumed that Rubin's claim that he had just returned from the war was another one of his lies/another part of his 'act.'

The only thing that might be slightly helpful (depending on what generation you belong to) is to understand the societal context of the Vietnam War, and the counterculture's attempts to avoid serving in it. If you know that bit of history already, then no, you probably don't need to watch
Greetings
first, though you should definitely watch it afterwards.
Greetings
makes clear that John Rubin was indeed a veteran who had just returned from serving in the Vietnam War, which means he wasn't just faking it at the end of
Hi, Mom!
when he was dressed like a soldier. This gives us the insight that perhaps his radical behavior in the sequel was related to his experiences in Vietnam.
Greetings
also shows us the beginnings of Rubin's obsession with "peep art", and we do meet Mr. Banner before he became a big-time dirty movie producer (though it's odd neither one recognizes each other in the sequel).

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No you don't, this is a sequel but only because Deniro plays the same character.