How hard is it to get an academic position in Australia/New Zealand with a PhD from the US?
I know many academics in Australia who obtained their PhD in the United States. In general, I think that overseas experience is seen as a positive, whether it be an Australian going overseas to do a PhD or a PostDoc or a non-Australian  obtaining their PhD or PostDoc overseas. Some ranking systems even rate departments based on the number of international faculty members.

Of course, you still need to meet the job requirements. Thus, it is still going to help if you have a great publication track record, have studied or worked in a world-renowned university with world-renowned academics, and so on. Most Australian universities have a job listing site where you can subscribe to job alerts. Also, you would generally need to have good English language skills.