I thought so as well - the lady said not to worry about it, though.
We tried to do that - they would not start him working without the SSN.
When he went back into SSA to request a new card and to have it sent to our PO box, the lady casually laid a piece of paper on the desk, told him "I'm not allowed to give you this, but I have to go get a piece of paper off the printer, so if you happen to see a number on this paper that I am absolutely not supposed to give you... so be it" and she walked away. He quickly jotted down his SSN, and when she came back she said "Yes, I definitely cannot give you this paper" and shredded it with a smile and a nod. I don't know this woman, but I would love to send her flowers lol
We were able to get hubby working starting Tuesday night, and we are just waiting now for the replacement card to come. The lady said 7-10 days, so I'm watching our Informed Delivery like a hawk. Checked with USCIS and the green card is being produced as of Jan 29.