I have a sentence like this
There are four people in his family. They are his parents, his younger sister and him.
Is is correct if I substitute "he" for "him" in the sentence above?
In short, the answer is no. But there is a longer explanation.
Consider the following from a purely syntactical point of view:
✓ I / she went to the store.
✗ Me / her went to the store.✓ My sister and I / she went to the store.
✗ My sister and me / her went to the store.✓ "I am / she is home!"
✗ "Me is / her is home!✓ "It is I / she!"
✗ "It is me / her.
In technically formal and traditional English, the correct answer to "Who is it?" is "It's I" or "It's she."
However, idiomatically, that's not what many people say.
To be understood in practice, the correct answer is "It's me" or "It's her."
Also consider the following from a purely syntactical point of view:
✓ He is in his family.
✗ His is in his family.
✗ Him is in his family.✓ He and his parents are in his family.
✗ His and his parents are in his family.
✗ Him and his parents are in his family.✓ His parents and he are in his family.
✗ His parents and his are in his family.
✗ His parents and him are in his family.
In technically formal and traditional English, the correct construction should be "His parents and he are in his family."
However, that's completely wrong from an idiomatic point of view. Nobody would ever say that and think it was correct. (They would say "His parents and him are in his family.")
Now let's look at your actual example:
There are four people in his family. They are his parents, his younger sister and him.
Can you replace him with he and be correct?
Only in a strictly formal and traditional sense. In fact, from that specific perspective, it would be the only correct word to use.
But, idiomatically, it would be wrong. It's not what's commonly used, nor what would be understood by most people. And modern guidelines of usage have shifted to accommodate the new usage.
So again, you would not want to do that.
"he"?? No, it is absolutely wrong.
"Case" defines the job the pronoun is doing in the sentence. If the pronoun is the subject of the sentence (or of a subordinate clause in the sentence), it's in the subjective case: I, he, she, we, they, who.
If the pronoun receives the action of a transitive verb, it's the object of the verb—so we're in the objective case: me, him, her, us, them, whom. For example:
He (subject) kissed her (object).
She (subject) slapped him (object).
Notice that a couple of pronouns are the same in both cases: you and it. ("Ye" used to be both a plural of "you" and the objective form of the pronoun:
"I gave ye a gift."
But those usages have been obsolete for centuries.)
Pronouns can also be objects of prepositions, words like by, from, through, after, apart, etc. So we use the objective case here as well: The poem was written by him. The gift came from her.
(via Ask the English Teacher)
So in your example, we should write him or:
"They are his parents, his younger sister & he himself"