Does the potential buyer of the business have the right to get my SSN

The answer to the first question is probably yes. If you are to be an employee post acquisition, they need your SSN for proper tax withholding and payment. Even if you quit today, they still need it to generate a W-2.

The answer to the second question is it depends what you mean by "give out." Can they sell your SSN for marketing reasons? No. Acquisitions can be structured differently such as a merger or sale of assets. In a merger, one person becomes the other, so in that case they are not "giving out" anything. With a sale of assets, your SSN is really needed to continue employing you and or allowing the acquiror to comply with tax laws.