This study investigates the set stability of switched Boolean networks (BNs) with arbitrary switching signals, based on invariant subsets. Set stability determines whether a BN converges to a given subset, while arbitrary switching signals can be used to characterize uncertainties and disturbances. An algorithm is proposed to calculate the largest invariant subset contained in any given set, as this is the key step in establishing the set stability criterion. Based on this algorithm, A necessary and sufficient condition for set stability is obtained. The result is then used to solve the robust output synchronization problem and the robust node synchronization problem for BNs in the presence of uncertainties and disturbances. Examples are presented to demonstrate the use of the proposed approach.