He's definitely a citizen, having been born in Louisiana, but he may or may not be a "natural born citizen". The only two instances where that matters, are for eligibility to be President or Vice President. It's not problem for him as governor, nor later would it be a problem should he choose to run for Congress, either the House or the Senate. He could have become a military officer (you have to be a citizen, but naturalized or native born is fine) and risen to Chairman of the Joint Chiefs, and that would be OK under the law and the Constitution.
The bone of contention seems to be defining “natural born”. As far as I am concerned if you are born here that fits the definition.
I would hate to see the courts clogged with the same foolishness should Bobby Jindal become Pres.