"No Person except a natural born Citizen, or a Citizen of the United States at the time of the Adoption of this Constitution, shall be eligible to the Office of President; neither shall any person be eligible to that Office who shall not have attained to the Age of thirty five Years, and been fourteen Years a Resident within the United States." US Constitution
It is not a legal bar for office as president of the US for someone to be a dual, triple, or even quadruple citizen of various countries. The possession of passports from other countries is also not a legal bar to office.
To be a dual, etc. citizen would undoubtedly cause many nationalist Americans to abstain from voting for a particular candidate. IMO that is why Cruz dumped his Canadian citizenship 18th months ago.
IMO the US Constitution should be amended to remove the "natural born citizen" language. This language bars naturalized Americans from the presidency no matter how long they have been citizens and no matter what their service to the country may have been. Two examples of such people would be Jennifer Granholm, the former governor of Michigan (brought to the US as a baby), and Arnie Schwarzenegger, the former governor of California.
This restriction on the office was understandable in the beginning of the Republic when the possibility of counter-revolution led from abroad was real but the restriction no longer is sensible. pl