Birth citizenship of U.S. presidental possibilities

I know that to be president of the U.S., you have to be born a United States Citizen. Does that mean that the Vice President has to be born a U.S. citizen, too, just in case he becomes president? And what about the others down the line? Like the speaker of the house?

It’s all in Amendments 12 and 20. Because of the 12th Amendment, the Vice President must be eligible to be President. However, the 20th Amendment, which details presidental succession, has no such clause, so the Speaker of the House, Secretary of State (Kissinger and Albright, for example), and so on can be naturalized citizens but will be skipped over if the line of succession gets to that position.

From the 12th Amendment:

“no person constitutionally ineligible to the office of President shall be eligible to that of Vice-President of the United States.”