@Lindsay ********
AFAIK there are no special rules for foreign kids born here, they need visas or they can overstay just like foreign kids that come in with their parents.
Actually children don't need a visa to exit and return (even if they're on overstay). Kids can get back in on a visa exempt entry (if they come from a country that gets those) overstay, leave and come back again and again.
As I said I know kids whove had multi-year overstays, leave and come back on a visa exempt entry and then overstay again.
Until they hit 15 there's no fine, and no negative thing at all for either the child or the parent.
I agree with you though Lindsay, that they SHOULD get the correct visa/extension for the child, and their employer KNOW this too.