The correct phrase is "He and I are good friends."
In this sentence, "He and I" are the subjects of the sentence, and "are" is the linking verb connecting the subject to the complement "good friends." When referring to yourself and another person as subjects in a sentence, you should use "I" as the pronoun, not "me."
Using "me" in this context would be grammatically incorrect. The pronoun "me" is used as an object, not a subject. For example, you would say, "He is friends with me," where "me" is the object of the preposition "with."
So, the correct version of the sentence is: "He and I are good friends."