It is easy enough to find eggs in cardboard containers. These can either be put out with your paper recycling, burnt, or when shredded, put in with the compost or worm farm.
Thanks to Andy for letting us know that the Bin Inn is quite happy for your to bring any empty egg containers you might have to be used by yourself and other customers to buy eggs.
Another alternative is to consider keeping your own chickens. Generally in NZ there are no restrictions in keeping chooks as long as you don’t have any roosters. They are great waste disposal units, as well as handy rotary hoes when preparing garden beds! We’ve always had at least a couple at a time and found that this way they have been able to defend themselves against cats no problem at all.