I'd agreed on that - 'Me and him' being 'incorrect'. I specifically meant to regard as acceptable (but informal) 'Him and I'.
Applying 5jj's analysis, 'him' is not objective or accusative or whatever you prefer, it's emphatic - like lui (in the emphatic use of that word, not the indirect object).
I don't think I've ever heard anyone say "Him and I," (as a sentence's subject) which is why I commented. It's certainly incorrect grammatically, but I don't think it's natural at all.