The first time you will be "required" to submit your SSN is when you enter Phase II and you register with GIS, the background check company. At the face-to-face interview, I believe I had to type it into their program along with my permanent address and DOB for my fingerprinting/DHS application. I think that's separate from your actual background check, not 100% sure on that one...

To answer your question- provide it, don't provide it. Don't think it matters on your app as far as selection goes... but you will have to provide it at some point in the process.