Be careful about this. The only time your entry into Asia doesn't count is when you're doing a transit between SWP and Europe, not SWP and North America. So you can do your AKL-HKG-JFK but then when you go Europe-Asia-SWP, you can't have a stopover in Asia.

BTW, I'm curious if anyone has tried something like this before. The rules imply that when you make your Europe-Asia-SWP flights as your second entry to Asia, you should keep right on going and spend < 24 hours in Asia. But they don't actually say that. It seems that you can spend 3 days flying all around Asia as long as you don't have a stopover anywhere in Asia. Is this true?