In order to find saver availability, you'll most likely have to route through Asia.

OZ files a respectable -- not great, but not terrible -- business class product from ICN to SYD.

If you have the miles and the time, TG F frequently has great availability from BKK to SYD. You can also fly TG J from BKK to AKL, but availability is usually not as good.

Connecting through NRT is a challenge, as ANA doesn't fly to SYD or AKL; NZ does fly to NRT, and occasionally an NRT-CHC leg shows up with decent availability. I believe you can occasionally find NZ J availability on PVG-AKL as well.

Remember that if you are redeeming from North America to Australia/New Zealand with MileagePlus, you aren't allowed to connect through Europe (otherwise FRA-BKK-SYD in F would be very tempting). You must cross the Pacific. No, this rule isn't published, but it seems to be consistently enforced.