Earlier quoted context omitted.
At the very least you need to provide proof of your SSN before you start employment in the US (as a US citizen at least). This is required for your social security benefits and some tax info. I believe in most states you actually have to physically show your social security card on the first day of employment now too.
I think you are thinking of form I-9, which is federally required. SS Card is one of may accepted forms of ID, I always use my passport instead.
Using your passport doesn't obviate the requirement that they have your SSN or ITIN for the IRS.
The SSN card proves your eligibility to work, they still need to verify your ID separately (DL, voter card, etc.). They need both pieces. A passport counts as both.