No. If your domain is the union of two externally tangent balls or the difference of two internally tangent balls, then it is not $C^{1,1}$. Not even Lipschitz. The boundary is not a graph of a function near the point where the balls are tangent. Perhaps you incorrectly formulated the question.

Edit: This is the answer to the previous version of the problem, before it was reformulated.

No. If your domain is the union of two externally tangent balls or the difference of two internally tangent balls, then it is not $C^{1,1}$. Not even Lipschitz. The boundary is not a graph of a function near the point where the balls are tangent. Perhaps you incorrectly formulated the question.