Clearly FALSE, as evidenced by the rulings of U.S. Secretaries of State.
Secretary of State Frederick Frelinghuysen ruled in 1883 that Ludwig Hausding, though born in the U.S., was not born a U.S. citizen because he was subject to a foreign power at birth having been born to a Saxon subject transient alien father.
Similarly, in 1885, Secretary of State Thomas Bayard determined Richard Greisser, though born in Ohio, was not born a U.S. citizen because Greisser's father, too, was a transient alien, a German subject at the time of Greisser's birth. Bayard specifically stated that Greisser was at birth 'subject to a foreign power,' therefore not "subject to the jurisdiction of the United States" within the meaning of the Fourteenth Amendment.
Hmmm...quoting Secretary of States, or the US Supreme Court...who gets to say what the Constitution means?
Further, I learned a long time ago that birthers parrot supposed cases without ever learning the facts. Unhappily, a search of “Ludwig Hausding” turns up a lot of birther parroting. Both of those examples seem to come from “A Digest of the International Law of the United States”, and both seem to involve people living in foreign countries as citizens of those foreign countries, who then claimed US citizenship to avoid military service.
If you have a fuller treatment of their cases, and in particular a court case involving them, please cough it up.