Nonono, she's right, and this is why: I was predicating my original argument on the supposition that in a comparision such as "They are not like him and me," that "him and me" were the objects of a verb. But they aren't. "Like/notlike" and "as" are not verbs. They're comparative conjunctions linking two independent clauses, each of which contains its own subject and verb (although in the second clause, the verb is usually ellided): "They are" and "Him and me are." "Him and me" is the subject of that second independent clause, and both clauses together make up the compound subject of the sentence as a whole, and so should be "He and I are." The verb in the sentence is "to be," and there is no object.

Clear as mud?