Triple Citizen
Registered Users (C)
Airlines at check-in time are only concerned about the passenger's ability to enter the country of destination. Even if they question the OP about lack of I-94/GC, the OP simply has to state the fact that he/she is a US citizen. When I check in for a flight to Toronto at O'Hare, I present my Canadian passport. I have never been asked about my lack of I-94 or GC.
Since you'll surrender the greencard at the time of your oath I dont see how you can board the flight.