That is incorrect. A person is not eligible to sponsor a parent until they're 18 (or 21?). So while a mother running over the border to have a child does benefit from better medical care (most likely), the child's citizenship doesn't help her for a long time, if ever. (The child would need to meet all the sponsorship requirements for parents, which aren't trivial.)

At best, this helps the child's future opportunities, which may help the family. It in no way gives the parents a pass. A cursory review of sponsorship laws would reveal this. Yet for some reason, people parrot this silly saying as if it was somehow true.

Sure: http://www.law.cornell.edu/cfr/text/8/101.3

The relevant portion is: "A person born in the United States to a foreign diplomatic officer accredited to the United States, as a matter of international law, is not subject to the jurisdiction of the United States. That person is not a United States citizen under the Fourteenth Amendment to the Constitution. Such a person may be considered a lawful permanent resident at birth."