Do you know if someone of U.S. citizenship must show a SS# to open an offshore bank account?
No. I can only tell you rules as apply to Australia. It's a 'numbers' system. From memory, your ID has to add up to something like 120. You get so many points for a passport, so many points for a Medibank Card, etc. You couldn't open an Aussie Bank Account with a US SS number. I don't think our Banks would recognize it.
PS Try the Bahamas!