No, I'm only an Australian citizen. However, I am able to serve in the British Army since Australia is part of the Commonwealth and all citizens in the Commonwealth are allowed to serve in the British Army without any of them having any obligation to become a British citizen in the end. if I remember correctly. In all, there are 7 000 soldiers in the British Army who are from the Commonwealth and who do not have British citizenship.
Citizenship requirements - British Army Website
The British Army looks attractive since the opportunities and careers available in the British Army are much wider and diverse than offered in the ADF. Also, they will have a Parachute Regiment for some time to come whereas Australia will soon abandon its parachute battalion (3RAR) in 2011 for a light infantry battalion. Bloody government.
For the US Army, yes, you do need a Green Card, however to become an officer you must be a citizen of the United States. Also, I think going in enlisted in the US Army, it is expected of you to become a US citizen in the end. I'm not sure though. An Australian captain from 1st Commando whom I met at cadets served in the 3rd Marine Division for 3 years and still has his Australian citizenship after coming back home half a year ago.