I was in the Artillery back in '65-'68 with some fellas that probably weren't citizens at the time. They were just as good as cannon fodder as we were. Or, as otherwise known, Targets.
None of us asked if they were citizens.
It's becoming more prevalent. When I joined the Army 20 years ago I don't recall any non-native born soldiers in Basic, AIT, or my first duty station. Now I see it all the time. We have soldiers from Pakistan, Nigeria, Surinam, you name it. Most enlist as green card holders. I don't like it and I was just as shocked as you when I found out we do this. It reminds me of the waning days of the Roman Empire when the Roman Army relied on Gauls, Goths and Vandals to fill-out the ranks.