III-nitride alloys continue to drive advances in electronic and optoelectronic devices. Recently, boron-containing nitride alloys have been explored with the goal of expanding the range of applications. Using first-principles calculations with a hybrid functional, we study the electronic structure of wurtzite BGaN alloys. Strong bandgap bowing is observed, with a concentration-dependent bowing parameter. Due to the strong bandgap bowing, the fundamental bandgap in strain-free alloys is effectively unchanged for the lowest B concentrations. A crossover from a direct to an indirect bandgap occurs for B concentrations greater than 50%.