You have got it all wrong, it is us kiwi`s who head over to OZ in June to escape our crappy winter weather.
June will be very wet and very cold with possible snow and ice depending on exactly where in the SI you will be heading..
Not really the best time for Motorcycling.
I was in the bottom of the south last year at that time, mainly for the Brass Monkey Rally. It was -2 at 3:30pm on the Saturday and -11 overnight. Lots of snow and ice throughout the various Road Passes. You may get away with some areas but generally its a not the best time to ride. I would probably suggest top of the south island possibly but the main adventure roads like the Molesworth and Rainbow are closed due to winter anyway. Good Luck.