Pretty much every employer in the U.S. requires a social security number. Actually, EVERY employer (legal, that is) requires an employee's SS number to be on file. Your 'green card' number, also known as the 'alien registration number,' is required on the initial government paperwork. Once the employer establishes you can legally work in the U.S., this issue (green card or U.S. citizen) often does not show up again in paperwork, but the SS number shows up everywhere.
Hope this helps.
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A tyrant is feared in life and hated in death.