Well there it is in black and white from the SSA itself.

"The Social Security Act does not require a person to have a social security number to live and work in the United States, nor does it require an SSn simply for the purpose of having one."

They go on to use circular reasoning by citing "other laws" that
require businesses who are required to file with the IRS, without
stating what those requirements are.

Pretty evasive, in a clever sort of way...