1. Try to find them in the US Censuses. If you're successful, you have clues as to if they became citizens, and if so, when.The censuses might also tell you when they immigrated.
2. Try to get their citizenship papers, based on above info. That could give you clues toward their previous names, and could have the precise travel information - names, ship, and dates.
3. Try to get the birth record of any of their children, or their marriage record, if privacy laws allow. That might have different names.
We can give you details on any of the above searches if you need it, and possibly give more help if you can give us any other dates, locations, etc.
We like to help --.
Hilary