Learn Croatian: Njezin brat je sretan.

In Croatian, possessive words must match the gender of the thing or person being owned, not the owner. Since brat (brother) is a masculine word, we use the masculine form njezin, even though the brother belongs to a female.

Why is je the third word here? I learned that short verbs like je must be in the second position.

You are right that je is an unstressed word (a clitic) that usually takes the second position. However, it can come after the first complete phrase instead of just the first word. Here, Njezin brat acts as a single block of meaning, so je goes right after it.

How would this sentence change if we were talking about her sister instead of her brother?

The word for sister is sestra, which is feminine. Everything describing the sister must also become feminine to match. The sentence would become Njezina sestra je sretna. Notice how njezin changes to njezina, and sretan changes to sretna.

I sometimes see njen used for "her". Could I use that here instead?

Yes! njen is just a shorter version of njezin. Both are correct and mean the exact same thing. You will often hear people say Njen brat je sretan in everyday conversation.