This is not an issue that the GMAT would test, since the GMAT will never write in the first person ("I" or "me").

The correct usage depends on whether the expression is a subject or an object. Just as we use "I" for a subject (I bought a book) and "me" for an object (He bought a book for me), so we would use "My brother and I" as a subject and "my brother and me" as an object:
My brother and I bought books.
My father bought books for my brother and me.

"Me and my brother" is never correct.