"Needs' is plural. I don't have only one need, I have several needs. I need more than one thing, I need many things. Plural.
"Everyone's" means that it belongs to every one of us, i.e. each of us.
The needs (plural) of each of us (plural) is different. Everyone's needs is different. The needs (plural) of every one of us, is different. ….. are different.