Although there are many questions here which deal with grammatical number and plural subjects, this is the first time I've heard of this "rule". It deserves its own answer.
You are entirely correct. A plural subject requires a plural verb, especially when each "element" of that subject might be considered an individual.
The only time where an apparently plural subject takes a singular verb is when that subject is considered a single entity in its own right:
Fish and chips is the best meal ever devised.
It's possible that whoever told you that erroneous rule has come across a sentence like that and simply come up with a reason which fits it (fish and chips may both be extracted and the sentence becomes singular).
But it makes no sense: if you remove Martin or John from your sentence, you are only talking about the remaining person. Any verb in that sentence must be singular. To extrapolate that and say that because each element is singular, both together can be is frankly ridiculous. Your intuition that it is ridiculous is correct. If a subject consists of more than identifiable and separable element which are not taken together as a composite entity, it must be plural.