Their alien parents can be, however. And kids generally live where their parents want them to live.
If a 30 year old Jamaican woman (without US citizenship) is deported for being a drug dealer, the fact that she has a 4 year old kid born in the USA is not going to stop the deportation. Where the kid lives will be up to mom. She'll likely take the kid to Jamaica...or she could get a US friend or relative to take the kid in. If the kid goes to Jamaica, he/she would be in good company. There are thousands and thousands of born-in-the-USA US citizen children living abroad.
The “anchor baby” element really doesn't kick in until the child moves back to the USA, usually around high school age, and then later as a young adult, files a petition (IR-5) to bring the alien parents over, legally, as lawful permanent residents (green card holder). Of course there are visa ineligibilities for criminals, deportees etc. In fact in Jamaica Mom's case, she's out of luck since narcotics dealing is a permanent immigration bar.
So, when the illegals all become Jamaican drug dealers with new babies, then we have a problem.
When they are older parents and grandparents, not newer illegals who Gingrich want to support, then they likely will have kids and grandkids old enough to keep them here.
You and I both know they’ll not be denied a visitor’s visa to visit family anytime they want.