June in Australia and New Zealand is the middle of the winter. So it will be as cold as it gets there. How cold depends on where you are and because of the climate changes we're having, previous records aren't always that dependable. The only thing I can think of that would change as far as where you could go is if you were traveling independently and wanted to go to Kakadu, that is accessible in their winter.
What you do about clothes is, as always, dress in layers. I have found a water repellent jacket with a removable lining and a hood the best thing. Carry that on the plane; don't pack it. Same with your sturdiest shoes. And you might want some kind of boot at that time of year. A warm hat as well as one for the sun, gloves and scarf. You don't need that many different clothes, just follow the advice of the experts and stick to one basic color scheme. And remember, there are laundry facilities for the things you don't want to just rinse out.
With all that said, I have only been to Australia and New Zealand between October and March which is their summer, so I really can't advise as to the difference in comfort between summer and winter.