the US Visa waiver stuff is based upon citizenship, not place of birth.
In the application form it asks specifically for your
place of birth. I know as I am a forces brat born overseas to a serving RAF father, and have come up against this issue personally. I was born in what is today a friendly EU country and so the 'problem' is usually squared away quite easily, but had my father been based in a now not-so-friendly Middle East country, I imagine it might not be so simple.