We present a numerical relativity study of head-on collisions involving boson stars using initial data inspired by the Bowen-York initial data used to model black hole binaries with punctures. The initial data method preserves the simplicity of the Bowen-York approach, thus allowing incorporating information from the early, post-Newtonian inspiral phase in binary coalescences. We test the method on a single boson star with linear momentum. We present results from head-on collisions of boson stars as well as encounters of boson stars with black holes. In general, the results are consistent with previous studies, demonstrating the effectiveness of the initial data method. In particular, we show that boson star head-on collisions emit more energy in gravitational waves than the equivalent black hole binaries. On the other hand, head-on collisions of a boson star with a black hole radiate less than their black hole binary counterparts.