He and I is sometimes correct, but him and I is never correct.
Technically, you can use “he and I” when you need a subject pronoun that functions as part of a compound subject or a compound predicate nominative (e.g., “I invited Jason over to watch the game because he and I both love the Kansas City Chiefs”).
However, the plural subject pronoun “we” is more concise than “he and I” or “she and I” (e.g., “Jason left early because we both have school tomorrow”).
“Him and I” is incorrect because “him” is an object pronoun and “I” is a subject pronoun.
“Him and me” is technically correct when you need an object pronoun that functions as part of a compound indirect object, direct object, or object of a preposition (e.g., “When my brother and I were teenagers, Mom gave him and me a strict curfew“).
The plural object pronoun “us” means the same thing as “him and I,” and it’s more concise (e.g., “When my brother and I were little, Dad read to us every night“).
A QuillBot Grammar Check can help you spot errors like “him and I” and use object pronouns and subject pronouns correctly.