What business do you run?
I deal with this every time I'm hiring. Fully half the applicants I get want to be paid in cash, when I tell them it's straight payroll they say "I can't do that". Not because they want the extra money, but because they don't have an SSN. I had one hire that was flagged by the SSA nearly a year after being hired. He was using a fake SSN. I had to release him or face a huge fine.
In construction paying in cash is common, and very profitable. The employee gets more money, the employer doesn't have to pay payroll taxes, doesn't have to pay overtime, doesn't have to pay workmens comp, and saves on general lability through reduced payroll. If you hire illegal's, you don't even have to worry about them getting hurt on the job. They won't report an injury for fear of being deported. I know of people that lost fingers and had broken bones that quit for fear of deportation.