Even with a green card you may well run into problems....look at the last line of the following paragraph that was taken (cut/paste) from a job notification...that was looking for a corporate co-pilot to fly company personnel on company aircraft..
SKILLS: Required: 1,000 hours flight time. 500 hours as Pilot in Command. Must hold an FAA Commercial Pilot certificate, Instrument Rating, appropriate Airplane and Multi-Engine Land Class rating. Ability to maintain licensure a must. Must hold and maintain FAA First Class Medical Certificate. Must be able to prove U.S. citizenship in order to obtain and maintain a U.S. Department of Defense Security Clearance.