Solution 1:

Everyone is a pronoun and means every person or all people. In your example everyone's denotes the possessive form of everyone. Remember that in AmE, it's always singular not plural. So the word following it should always be singular not plural, too. Here are a few examples:

He got everyone's attention.

Everyone needs him. (not need)

Everyone likes him. (not like)

Everyone's mom was there. (not moms)

That said, everyone's is also the abbreviated form of everyone is.

Everyone's home for dinner means Everyone is home...