The Arizona legislature does not get to define what a natural-born citizen is.
They don't have to, the US Government already has over 150 years ago:
NY Times:
THE NATURALIZATION QUESTION.; THE POSITION OF THE ADMINISTRATION DEFINED. Important Official Paper.
http://select.nytimes.com/gst/...
THE NATURALIZATION QUESTION.; Attorney-General Black's Opinion upon Expatriation and Naturalization.
http://select.nytimes.com/gst/...
[H]ere none but a native can be President…;A Native and a Naturalized American can go forth with equal security over every sea and through every land under heaven, including the country in which the latter was born…They are both of them American citizens, and their exclusive allegiance is due to the government of the United States…One of them never did owe fealty elsewhere, and the other, at the time of his naturalization, solemnly and rightfully, in pursuance of public law and municipal regulations, threw off, renounced and abjured forever all allegiance to every foreign prince, potentate, State and sovereignty whatever, and especially to that sovereign whose subject he had previously been. [end quote]
Allegiance to but one nation, either at birth or by naturalization. Period. Here endeth the history lesson