I'm grateful to Andy Porter for pointing out an interesting article in The Globe and Mail of Toronto on the complicated tax status of U.S. nationals living abroad.
Basically, Americans citizens are generally required to file U.S. income tax returns no matter where they earn their income and no matter where they live. They do get credit for foreign taxes paid, so they may not owe much or any U.S. tax, but it's at minimum a hassle.
And this applies even to citizens who have never actually resided in the U.S., such as children born to U.S. citizens abroad and to those born to foreign mothers who are just visiting the U.S. It also applies to those who have citizenship in another country, either because by birth or naturalization.
Though the article doesn't mention it explicitly, the U.S. Supreme Court has ruled in at least two cases (Afroyim v. Rusk and Vance v. Terrazas) that under the Constitution, U.S. citizens do not lose their American citizenship except by explicitly renouncing it. Becoming a citizen of another country doesn't count.
And as the article does note, formally renouncing U.S. citizenship isn't easy. There's a form to fill out and fairly sizable fee to pay, and even worse, renunciation can result in a huge tax bill because of of the treatment of capital assets such as homes.