If the Constitution doesn't forbid it (and it doesn't) AND the States chuse 270 electors for President of the United States who vote for that person, of course he/she can.
You are correct.
But can a dual citizen get the security clearance for that position? I don’t think the DoD would allow it. They don’t even allow passports holders of other nations to have that clearance of lower level clearance.
http://www.dod.mil/dodgc/doha/isp.html