The quote 'Come on you Apes, do you want to live forever?' is generally attributed (it's possible other versions were uttered before) to Dan Daly, a Marine. Heinlein's military service in the Navy would push me towards the Marine Corps as the influence. As far as I know Heinlein never said if the Mobile Infantry is based on any specific branch.
This is going to be anecdotal, but I would say at this point it's impossible to draw a conclusion.
