Correct me if I'm wrong, but I believe that "eligibility" to anything is determined by enacted laws and regulations to define your eligibility. Therefore, if you want to talk about "eligibility" to a position that requires one to be a U.S. "citizen", you have to talk about what are the 'laws and regulations' in being a United States "citizen".
The Constitution refers to an individual having to be a "citizen". What kind of citizen? A "natural born Citizen".
You can only become a U.S. "citizen" by either 1 of 2 ways. By "birth" and by "naturalization", which currently provides for the 2 legal citizenship status in this country.
Show me any law or regulation with respect to being a "U.S. Citizen" where you have 3 classes of U.S. Citizenship where you don't fit into the first 2?
Those who want to 'define' a separate distinction for those being BORN HERE as being a "Natural Born Citizen", versus those being BORN HERE but not being a "Natural Born Citizen", are creating a third type of citizenship status that is not defined currently under any legal protection.
Although the headline of the FR link claims otherwise, there is a legitimate dispute about the phrase “natural-born citizen”:
http://www.freerepublic.com/focus/f-chat/2785458/posts
The Federalist Society has had at least one debate on the subject.
Files of case law for the side Obama doesn’t like have disappeared from the Justia records.
Whichever side is right, is does not make sense for Republicans and others to bring this dispute into a very important election—to their detriment.