lets get real here
Iron Maiden,
Yeah, you need a green card to work here. Don't take it out of context but it really doesn't matter what country you come from, as long as ur not Muslim you wont have any trouble getting a job here. We pretend to treat everyone as equals but you'll figure out shortly after you get here that's a big load of bum scoop.
But otherwise most US airlines gonna be pretty desperate for pilots in the next year or so. With a green card ur pretty much in the club and u can apply anywhere you want to (best of luck to you). I think your Aussie accent will get you far though, working here just depends on how much initiative you want to take in doing the research & filling out the visa apps.
Here's an idea for ya, I've been wrong before so do your research but I think guys at Cathay Pacific can take bases anywhere in the world. As long as you've got the right to live/work there, you can take the base...Cough,,cough *Green Card* Cathay has bases both in Aussie as well as the USA obviously so u might consider that route too.
I don't know what the situation is over in your part of the world now but I think I'd prolly stay in Aussie if I were you. That whole side of the world is going to take center stage in the coming years which arguably implies job security for you. But I thought it was real funny that you posted that question on Independence Day (4th of July) hahaha!
Whatever you decide to do, Good luck...hope this insight helps out