You may be able to get an ITIN number from the IRS so you don't have to give out your SSN.
For reason, check 'other' and simply indicate that you do not want to disclose your SSN to your customers.
Once you get your 9 digit ITIN, that can be used as your payee ID number on the 1099 instead of your SSN.
I looked up "Federal Tax ID number" at the home page for the IRS: WWW.IRS.GOV and the ITIN starts out:
An ITIN, or Individual Taxpayer Identification Number, is a tax processing number only available for certain nonresident and resident aliens, their spouses, and dependents who cannot get a Social Security Number (SSN). It is a 9-digit number, beginning with the number "9", formatted like an SSN (NNN-NN-NNNN).
Don't know if this is really what you need or want.