Animals such as bees, ants, birds, fish, and others are able to perform complex coordinated tasks like foraging, nest-selection, flocking and escaping predators efficiently without centralized control or coordination. Conventionally, mimicking these behaviors with robots requires researchers to study actual behaviors, derive mathematical models, and implement these models as algorithms. We propose a distributed algorithm, Grammatical Evolution algorithm for Evolution of Swarm bEhaviors (GEESE), which uses genetic methods to generate collective behaviors for robot swarms. GEESE uses grammatical evolution to evolve a primitive set of human-provided rules into productive individual behaviors. The GEESE algorithm is evaluated in two different ways. First, GEESE is compared to state-of-the-art genetic algorithms on the canonical Santa Fe Trail problem. Results show that GEESE outperforms the state-of-the-art by (a) providing better solution quality given sufficient population size while (b) utilizing fewer evolutionary steps. Second, GEESE outperforms both a hand-coded and a Grammatical Evolution-generated solution on a collective swarm foraging task.