In this paper, the outage performance of downlink non-orthogonal multiple access (NOMA) is investigated for the case where each user feeds back only one bit of its channel state information (CSI) to the base station. Conventionally, opportunistic one-bit feedback has been used in fading broadcast channels to select only one user for transmission. In contrast, the considered NOMA scheme adopts superposition coding to serve all users simultaneously in order to improve user fairness. A closed-form expression for the common outage probability (COP) is derived, along with the optimal diversity gains under two types of power constraints. Particularly, it is demonstrated that the diversity gain under a long-term power constraint is twice as large as that under a short-term power constraint. Furthermore, we study dynamic power allocation optimization for minimizing the COP, based on one-bit CSI feedback. This problem is challenging, since the objective function is non-convex; however, under the short-term power constraint, we demonstrate that the original problem can be transformed into a set of convex problems. Under the long-term power constraint, an asymptotically optimal solution is obtained for high signal-to-noise ratio.