As a sole proprietor, am I required to provide my social security number to a customer who wants to issue me a 1099 Misc? I provided monthly swimming pool maintenance.

Yes, I'm afraid so.  If you don't like doing that, you can always apply for an EIN for your business.  You don't have to have one unless you have a payroll, but you can always get one, just in case you need it for this purpose.

You can also explain that your customer does not need to provide you with a 1099 unless she is a business and wants to use her payments to you as a business expense.