Federal law prohibits employment discrimination based on race, color, religion, sex, or national origin (see Title VII of the Civil Rights Act of 1964). There are some exceptions for U.S. Government positions that require special clearances. But generally, as long as you have a Green Card and therefore the ability to work in the USA they cannot discriminate against you on the basis of your national origin.

Whether they do in fact discriminate in practice, I cannot tell you.

P.S. Showing that they can't find a suitable candidate is a requirement to get work visa for someone who is otherwise not authorized to work in the USA – this absolutely does not apply to you as you have a Green Card.