But there are other ways for those children who are American citizens with illegal immigrant parents to stay here. They can stay with someone who is willing to take them in, such as another (legal) family member or a friend. I've known people who were willing to allow a teenager to stay with them for a year or two just to finish school because the parents were moving but the teen didn't want to leave their school. My relatives and a couple of my friends offered to allow me to stay with them when my family moved out of state right before my senior year of high school.
Honestly though, there shouldn't be citizenship based on where you are born. It should fall to what citizenship your parents have. For those here already, if they are old enough, they and their parents should arrange a place for them to live as I described above. If they are younger or no arrangements can be made, then they are still the responsibility of their parents. It is their parents' fault for trying to live illegally in a country, instead of either going through the legal methods of entering/staying in a country or trying to improve their own country.