I used to be an instructor on the CRJ and some of my best students were guys from Gulfstream. I'm not promoting the program, just presenting my own personal observations. We had a class come through training that was mostly made up of gulfstream guys and they all did exceptionally well in training. I found their instrument skills to be very good.
(And now back on topic)
There are many greencard holders working for US Airlines, there is no reason why an airline would give preference to a US Citizen over a permanent resident, I have worked at 3 airlines, all of whom hire foreign nationals. I'm not saying you won't run into prejudices and obstacles at some places but if you do just keep looking