Deterministic dynamical system models with delayed feedback and state constraints arise in a variety of applications in science and engineering. Under certain conditions oscillatory behavior has been observed and it is of interest to know when there are periodic solutions. Here we consider a prototype for such models --- a one-dimensional delay differential equation with non-negativity constraints. We obtain sufficient conditions for the existence of slowly oscillating periodic solutions of such equations when the delay/lag interval is long. Under further restrictions, including longer delay intervals, we prove uniqueness and uniform exponential asymptotic stability of such solutions.