Yes they will, but don't worry too much about them being "under the table". What you should do though is gather up the info of those you will put down as a reference and ask them if its okay to do so and of course, tell them people will be calling them to ask them about you. If you don't want them to disclose that you were a "cash" employee, tell them so, but they usually don't ask that. They just ask them things like if you were a good worker, on time, motivated, etc.

You should have a minimum of 3 professional references ready, preferably that of your supervisors, and 3 personal references, people you've known a long time.

As far as the background check, I'm not sure how that works if you've never had a ssn. Usually they make you sign a form with your name & info, including DOB, SSN, and a driver's license.