A classical theorem in number theory due to Euler states that a positive integer z can be written as the sum of two squares if and only if all prime factors q of z, with q ≡ 3 (mod 4), occur with even exponent in the prime factorization of z. One can consider a minor variation of this theorem by not allowing the use of zero as a summand in the representation of z as the sum of two squares. Viewing each of these questions in Z_n, the ring of integers modulo n, we give a characterization of all integers n ≥ 2 such that every z ∈ Z_n can be written as the sum of two squares in Z_n.