IIRC, you cannot hold a position in a foreign government, and retain your US citizenship.
Well, sort of. According to the
State Department website, if you hold a non-policy level position in a foreign government, the State Department will presume that you don't intend to renounce your citizenship. But if you voluntarily hold a policy level position (like being a Member of Parliament), the presumption is that you intended to renounce your citizenship. I don't think you officially lose your citizenship, though, unless you file the necessary paperwork with an overseas consulate or some sort of administrative action is brought against you.
The United States is hardly the only country that requires its citizens to enter and leave the country using that country's passport, by the way.
Most U.S. airports don't have sterile international transit areas, like in Europe. Thus, people transiting from Europe to Mexico could theoretically leave the airport while en-route. Houston's new international terminals might be able to be set up as a sterile area, but currently isn't, since ICE regulations require visas for transit through a sterile area anyway.
My impression and I may be mistaken is that Boris has previously entered and departed the US using a British issued passport and that this requirement is relatively new, (most likely about five years old) the fact that he can renounce his US citizenship and have apparently NO problem entering makes the rule and my understanding of it at this time fairly ridiculous..