"He were" and "she were" are examples of English subjunctive. This form most commonly shows up after "if" or the verb "wish." It basically expresses hypothetical situations. Some examples: He wishes he were a bit skinnier. If she were taller, she'd be able to reach the top shelf.

When you use "is/was" with "he/she/it", you are talking about real events. Things that actually happened. When you use "were" with "he/she/it" (or even with "I"), you are talking about imagined events. This is the subjunctive form. The word "were" (waere) was actually used for the subjunctive long before it was connected to the past tense plural. There's a lot of overlap in the verb-form between subjunctive and indicative moods, but they are not the same thing.

I used a subjunctive-mooded sentence. Did you notice that?

Arthur, subjunctive mood is very difficult to learn and use. I suggest, most learners of English avoid using it in English, but be aware of the existence of it. We should understand the implied meaning when native speakers are using it, and check your writing if any of your sentences actually are subjunctive-mooded and have implied meaning that you yourself do not want to express.

I can't add to what people have already said. I would have just given examples as I only teach by examples, "if he were to try and knock Mike Tyson out, he'd have a really bad day!" haha "If she were to try and take over this company, I think she'd succeed and manage it better". It's like suggesting something, If **** were **** to ****** then something would happen.