The only pertinent information to establish native born citizen status is one’s place of birth within the United States of America.
Mexican-American anchor babies whose parents are Mexican nationals and in the US illegally are natural born US citizens if they are born on this side of the border.
If Barack Obama was born on Waikiki Beach and delivered by a midwife, he’s a Natural Born Citizen.
Under the US Code, the only children born in the US who aren’t Natural Born Citizens are the children of foreign ambassadors and foreign embassy staff.
Title 8/Section 1401 of the US Code defines “Nationals and Citizens of the United States at birth.”
http://www.law.cornell.edu/uscode/8/1401.html
“and subject to the jurisdiction thereof” does not include people here illegally. Jurisdiction is not just about geography.