I think the whole sentence sounds off to begin with. How can one "go" with an assignment in the first place? Perhaps I'm just not familiar with the expression.
Everyone in this context is a second person pronoun, and it doesn't make sense to speak to someone (or a group of someone) in the second person and then refer to them with third person pronouns (his/her).
Perhaps one of these alternatives:
How is everyone doing with the assignment?
How is everyone's assignment going?
How is everyone doing with your assignments?
Note that with the last example, the actual second person point of view is consistent throughout.
The confusion comes from the fact that everyone can be either a third or second-person pronoun. You're using it as a second-person pronoun in this case, but then trying to follow it up with third person pronouns which you can't do.