Can someone from American Samoa run for President?

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The "natural born" clause means that if they were born on U.S. owned soil, they're eligible for the Presidency. Any territory that is American controlled at the time of that person's birth would make said person eligible in terms of the "natural born" clause.

"Natural-born citizen" includes an American born to citizen parents who recognize the child as theirs. One citizen parent would be adequate (thus even if Barack Obama had been born in Kenya he would be eligible for the Presidency). 

Citation needed.  This isn't how the Naturalization Act of 1790 defined "natural born".

It used the fairly basic definition of "natural born" equals "citizen since birth".  If Obama's parents had been living in Kenya when he was born, then he would not have been a natural born citizen, as children born abroad with only one citizen parent only get automatic citizenship if they come to live in the US while they are still minors, but that citizenship is not retroactive to birth.

Are you saying Ted Cruz is ineligible?

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Barry Goldwater was born in Arizona when it wasn't a state. He ran in 1964 and would have won if millions and millions of people had voted differently.  Wouldn't it be the same thing?

It was an incorporated area of the United States, so he had full citizenship at birth, regardless of his parent's status. The thing is none of the territories today are incorporated, except for Palmyra.

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But, what if, they got American citizenship later (like Tulsi Gabbard)? Would they be considered naturalized and thus ineligible for the Presidency?

Tulsi Gabbard's father was a US citizen, so it doesn't matter where she was born, she was a citizen at birth.