Employers are required to verify that the employee is eligible to work in the U.S., meaning they're a citizen or else have the proper kind of visa, and are not

Employers are required to verify that the employee is eligible to work in the U.S., meaning they're a citizen or else have the proper kind of visa, and are not in the country illegally. And most companies will also do background checks. So yeah, you'll need to give your name, and probably a Xerox of your passport or driver's license, when you sign on to a company.