Those who gain citizenship in another country ought to be stripped of their American citizenship.
With some exceptions, I agree. The gaining of citizenship in another country must require an act of volition on the part of the person for it to affect his or her US citizenship. I say this for two reasons:
1) If I recall correctly, some countries (I think Germany is one) automagically grant citizenship to anyone who is born in their borders, even if "within their borders" means "on a foreign military base within their borders". So a kid born to military parents stationed on a base in one of those countries would have dual citizenship through no act of volition. Perhaps people wishing to run for president could be required to renounce such dual citizenship, just to be safe.
2) We cannot control another country's laws with regard to who is and is not a citizen of that country. Suppose a country decides that Candidate A in the US presidential election is
much better for their interests than Candidate B. >poof< New law saying that the country can declare anyone they want a citizen. They issue Candidate B a passport without his knowledge or consent, and, oops, suddenly Candidate B is a stripped of US citizenship, ineligible for election for President.
Anyone who
seeks citizenship from another country, or who makes use of the benefits of citizenship in that country (for instance, traveling under a passport from that country), yes.
-BP