Have you asked the flight school that's offering you the job if they would be willing to sponsor you for a green card? (I'm sure they'd have to prove that there weren't any American candidates which I would expect to be a little tricky...)
A company would not be able to sponsor him/her for a green card, but rather a different type of visa, most likely an H1B visa, but I have not heard of any pilots getting these recently. The J and F visas are the ones which flight schools are issuing, which have the ability to stay and work as a flight instructor for a limited amount of time. A search on these forums will give you some details about the J1 and F1 visas. Also, search for "OPT" and "CPT".