Persons in the country with out permission.

What ever label you decide to use, if you are in the United States of America without permission, you have committed an offense against the law of the land. That wasn't so hard was it.

As for as children who are natural born citizens of this country but their parents are not in the country with permission then the children will continue to be citizens, with all of the protections and benefits of citizenship, where ever their parents are. If that is out of this country so be it. I would process the parents as being in the country without permissions and they of course take their citizen child with them when they leave. If the do not want to take their citizen child with them then we have a process to care for and protect children without parents. Kind of like mom and dad are killed in an accident. We know how to deal with that tragic situation.