“May a third party debt ask my employer for my social security number?”
Sure, they may ask all they want, but your employer would be violating any number of legal obligations if they gave out any information about you. An employer may generally confirm whether you currently or previously worked at a company, but HR departments seldom answer anything beyond that, because it’s asking for trouble. Providing any other information about an employee, unless they are under some legal obligation to do so (such as for tax reporting, insurance/worker’s comp claims, responding to lawsuits, etc.), runs afouls of many state employee protection laws, federal laws relating to financial and healthcare privacy, and may also be a violation of the commitments made in documents such as employee handbooks.
If you have reason to believe your employer has been giving out personal information about you, you should contact an attorney in your area who specializes in employee rights matters. Contact your local county or state Bar Association and ask them for a referral to someone skilled in your area.