I probably shouldn't be wandering into the messy place that is politics, certainly not American politics anyway, but I find it surprising that in an immigrant country - a country whose strength has always been its willingness to let people from all over the world come and contribute - that an idea should be so prevalent that citizens-by-birth are by definition more 'American' than citizens who come by it later. Obviously a presidential candidate should be a citizen. That goes without saying. But people who become citizens later should be accepted unconditionally on all levels with all other citizens - otherwise, you send your country down a path into two-tiered citizenship, and as the country that pretty much has the patent on 'all citizens are created equal', that's very sad.
Obviously, a president should know the country and 'exemplify' it in one way or another. But I think it's been said that the people as a whole probably wouldn't vote for someone they saw as 'foreign' anyway - look at the current Birther movement.
It's an interesting contrast - in Canada the last foreign-born Prime Minister we had was in the 1980s - hardly ancient history, and at the moment we have one American-born PM candidate and one PM candidate who spent most of his adult life in the USA and the UK. While in both cases people discuss their electability as a result of these things, I've never heard anyone say they shouldn't be allowed to run.