All airline applicants must provide a 10 background history. I'm not sure if a US airline would be willing or able to verify the history of someone who has not been in the US for 10 years. This rule came about just after 9/11.
On the subject of pilot jobs and greencards, it is against US law for someone to be refused employment on the basis of having a greencard only. Either you are legal to work or you are not
AL