In this paper, we study the existence of solutions for nonlinear problems with piecewise constant arguments and reflection, focusing specifically on a first-order problem with periodic boundary conditions. First, we revisit previous results that allow us to define the Green’s function of the problem, characterize its properties, and determine the regions of constant sign. Based on these findings, we apply various fixed-point theorems to ensure the existence—and, in some cases, the uniqueness and sign—of the solution under different assumptions. In particular, we make use of Krasnosel’skii expansion-contraction theorem, Schauder fixed-point theorem, and Krasnosel’skii fixed-point theorem.