Consider the following sentence:
"Even during the simple occurrence of him and me standing next to each other makes me notice that he's taller than me."
Is him and me correct? Should it be he and I?
The choice is not between him and me and he and I, but between him and me and his and my. But in any case, this sounds like a fabricated sentence unlikely to occur in the normal speech of native speakers. Apart from anything else, there's something wrong with the syntax. It looks as if you want the simple occurrence to be the subject of the sentence. If so, you can't have during there.