I don't think only native citizens, if you have the right to live and work in EU, you can still come over if i'm not wrong.. During the training you should get a visum and if you get work a green card or something. Not pretty shure but check this link and try to contact them, ask questions..
http://www.easyjet.com/EN/Jobs/Pilot...hipscheme.html
One of the requierments is;
Nationality: You must be able to prove your unrestricted legal right to live and work in the EU and have no reason to believe that you will not be accepted for training in New Zealand.
Good luck!
/niko