U.S. Nationals are not eligible to serve as President or Vice President of the United States of America. “Citizens of the United States at birth” are not eligible to serve as President or Vice President of the United States, unless they are also natural born citizens by virtue of being born in the United States with both parents being U.S. citizens at the time of birth to exclude any and all natural born, jus sanguinis and jus soli, allegiances to foreign sovereigns. U.S. Citizenship for a child born in a foreign jurisdiction must be requested from the U.S. Department of State by application to recognize the foreign birth, and such a U.S. Department of State recognition of U.S. citizenship by foreign birth is subject to legal revocation. Such a revocation of U.S. citizenship by the U.S. Department of State is not possible in the case of a natural born U.S. citizen, because the natural born allegiance to the United States resulting from having two U.S. citizen parents and birth within the United States precludes any possibility of foreign citizenship and allegiance at birth.
Where’d that come from?
The 14th Amendment defines citizenship this way: "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." But even this does not get specific enough. As usual, the Constitution provides the framework for the law, but it is the law that fills in the gaps. The Constitution authorizes the Congress to do create clarifying legislation in Section 5 of the 14th Amendment; the Constitution, in Article 1, Section 8, Clause 4, also allows the Congress to create law regarding naturalization, which includes citizenship.
Currently, Title 8 of the U.S. Code fills in the gaps left by the Constitution. Section 1401 defines the following as people who are "citizens of the United States at birth:"
Anyone born inside the United States *
Any Indian or Eskimo born in the United States, provided being a citizen of the U.S. does not impair the person's status as a citizen of the tribe
Any one born outside the United States, both of whose parents are citizens of the U.S., as long as one parent has lived in the U.S.
Any one born outside the United States, if one parent is a citizen and lived in the U.S. for at least one year and the other parent is a U.S. national
Any one born in a U.S. possession, if one parent is a citizen and lived in the U.S. for at least one year
Any one found in the U.S. under the age of five, whose parentage cannot be determined, as long as proof of non-citizenship is not provided by age 21
Any one born outside the United States, if one parent is an alien and as long as the other parent is a citizen of the U.S. who lived in the U.S. for at least five years (with military and diplomatic service included in this time)
A final, historical condition: a person born before 5/24/1934 of an alien father and a U.S. citizen mother who has lived in the U.S.
* There is an exception in the law — the person must be "subject to the jurisdiction" of the United States. This would exempt the child of a diplomat, for example, from this provision.
Anyone falling into these categories is considered natural-born, and is eligible to run for President or Vice President.