***** NOT A TEACHER *****
Martin Hewings would disagree with your "rule". Perhaps the following extract may shed some more light on this matter:
When a subject is made up of two or more items joined by (either)...or... or (neither)...nor... we use a singular verb if the last item is singular (although a plural verb is sometimes used in informal English), and a plural verb if the last item is plural:
- Either the station or the cinema is a good place to meet, (or ...are... in informal English)
- The President or his representatives are to attend the meeting.
If the last item is singular and previous item plural, we can use a singular or plural verb:
- Either the teachers or the principal is to blame for the accident. (or ...are to blame...)
(Advanced Grammar in Use
- Martin Hewings, p. 82)
I think that most native speakers who use the "XYZ and me" or "Me and XYZ" construction would recognize that "XYZ and I" is also possible, if not more correct. It just so happens that the wrong construction is used by speakers, despite being ungrammatical. I think that those who'd say something like "Me and my buddy went to the pub" know that ths construction is ungrammatical. Why? Well, no native speaker would say "Me went to the pub".