In BE, it doesn't really matter whether it's subject or object; many would use 'him' there. In fact, many speakers would never use the formal form at all.
True. True. I hear it, too. The standard, ahem, informal standard these days is to use a noun (e.g. Michael) or the object pronoun (i.e. him) instead of a possessive noun or pronoun (Micheal's / his). The evolution of such wonderful new forms has everything to do with that nasty little, Chomsky-lovin' preposition 'about'. It's a great example of Language in Change--right up their with It's me ~ It's I. Speakers, having lost the traditional form, rely on the general rule: use an object noun/pronoun after a preposition.
What's wild about that is this. Changing the rule changes the syntactic and associated semantic structure:
They (Subject) disgaree about (Verb) him (Direct object) putting up his posters (Indirect object).
They (Subject) disagree about (Verb) his putting up his posters (Object).