Would a person be ineligible to receive a work permit if they had used a fake social to work before?

Committing a crime in the U.S. may get someone deported. Whether you are eligible to receive a work permit if you used a fake social security number to work before is a separate question and some times people in that situation are still eligible to receive the work card. 

Unless someone has been convicted of a crime for the activities you describe, it would not be a bar to applying for and receiving prosecutorial discretion in the Deferred Action for Childhood Arrivals initiative (including obtaining a work permit).  A criminal record (conviction) would present a problem, but merely potential criminal activity or activity that is not part of a public record would not hinder one's ability to take advantage of this program.