I'm not convinced she could not have a Social Security Number even if she wasn't a citizen, but maybe others can cite the official rules on that question.  I went to look at the 1952 manifest mentioned.

The numbers associated with Murray and Helen on that 1952 passenger list look to me like Certificate of Naturalization (CN) or Naturalization Certificate (NC) numbers (which are the same thing).

My guess is Murray’s NC 2166347 would date from sometime in the 1920’s.

Helen’s NC 6517534 would date from right after WW II (ca 1944-46 or so, maybe a little later).

If you can find a court record of Helen’s naturalization and confirm she was issued that certificate number (it should be written on the back of her granted petition for naturalization or might appear on an index card), then you might want to request that C-file by number from USCIS (uscis.gov/genealogy).  The C-number is above 6.5 million, so it should be a substantial file with many documents showing place of birth—perhaps even her sworn testimony with that question.

Marian Smith