That's a non sequitur.
The Employment Eligibility Verification System would make you completely at the mercy of the Department of Homeland Security in order to be "eligible" to work. If they require biometric data from you, and you refuse, you cannot be employed anywhere in the United States. If they adopt the REAL ID card with a RFID chip, as is currently the plan, you cannot work without one. The potential abuses of that technology are readily available on the 'net for your information.
And no, per the Privacy Act of 1974, there are only twelve conditions under which you are required by law to produce your SSN, and applying for a job is not one of them.