In a recent article, Nowicki introduced the concept of a special number. Specifically, an integer d is called special if for every integer m there exist solutions in non-zero integers a, b, c to the equation a² + b² - dc² = m. In this article we investigate pairs of integers (n, d), with n ≥ 2, such that for every integer m there exist units a, b, and c in Z_n satisfying m ≡ a² + b² - dc² (mod n). By refining a recent result of Harrington, Jones, and Lamarche on representing integers as the sum of two non-zero squares in Z_n, we establish a complete characterization of all such pairs.