Does the sentence contain a pronoun agreement error? After anyone eats Yzma's famous chili, they swear to never try another kind.

Yes.  There is an agreement error between 'anyone' (a singular pronoun) and 'their'  (a plural pronoun).

Explanation:

Some ways to fix the error

After anyone eats Yzma's famous chili, he or she swears to never try another kind.

OR

After people eat Yzma's famous chili, they swear to never try another kind.

By the way, there is another error in the sentence

NO:   'to never try'

YES:   'never to try'

the significance of the title, "the passing", is related to the fact that the main characters, who are three african-american descendant women, are passing as white. this means that, because of their light-coloured skins they look like white women. one of the three friends, clare, is married to a white man and she lives in a white community and she has never told her husband that she is black. the title can easily be connected to the idea of pretending to be someone you are not and of losing your real identity in order to be accepted by society.