1. All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the State wherein they reside.
There has been some debate in the past as to the flexibility of the bolded line. Legally, for example, the 14th Amendment did NOT extend citizenship to Native Americans. They were excluded from the "jurisdiction" of the United States, and therefore weren't granted birthright citizenship. A constitutional challenge on that exclusion was attempted once, and the Supreme Court did affirm a Congressional right to define the jurisdiction required for the 14th Amendment. Native Americans didn't become U.S. citizens until Congress passed the Indian Citizenship Act, extending their jurisdiction to the tribes.
There has been a theoretical discussion, stretching back to the Reagan era, that Congress could potentially pass an act that declares undocumented immigrants to be outside of the jurisdiction of the United States (e.g., the U.S. will abstain from any legal authority over them), which would eliminate birthright citizenship for their children. There IS a very real, and very much unanswered, question about the constitutionality of doing so. U.S. Supreme Court rulings on birthright citizenship refer back to U.S. v. Wong (a person born in the U.S. of noncitizen parents), but in that case the parents were here legally, and there was no question of jurisdiction. There are currently other jurisdictional exclusions in place (children of foreign diplomats don't gain citizenship, as an example), and there has never been an actual Supreme Court ruling on the citizenship of a child born to parents who are here in violation of the law. There really is no consensus, and this may in fact survive challenge.
Hunter has apparently decided to test the theory. Writing it off out of hand may be a mistake, as should taking a "who cares, the Supreme Court will strike it down anyway" approach.