From here, the discussion typically progresses to Ted Cruz. So before you ask, yes, I believe Cruz is eligible. However, the courts have not ruled on it. See here for my logic on Cruz: http://www.freerepublic.com/focus/bloggers/3289769/posts?page=102#102
So it actually would not require amending the Constitution so much as having Congress tighten the definition of NBC as is in their purview if I’m reading you correctly?