HESS J0632+057 is an eccentric gamma-ray Be binary that produces non-thermal radio, X-rays, GeV, and very high-energy gamma rays. The non-thermal emission of HESS J0632+057 is modulated with the orbital period, with a dominant maximum before apastron passage. The nature of the compact object in HESS J0632+057 is not known, although it has been proposed to be a young pulsar as in PSR B1259-63, the only gamma-ray emitting high-mass binary known to host a non-accreting pulsar. In this Letter, we present hydrodynamical simulations of HESS J0632+057 in the context of a pulsar and a stellar wind interacting in an eccentric binary, and propose a scenario for the non-thermal phenomenology of the source. In this scenario, the non-thermal activity before and around apastron is linked to the accumulation of non-thermal particles in the vicinity of the binary, and the sudden drop of the emission before apastron is produced by the disruption of the two-wind interaction structure, allowing these particles to efficiently escape. In addition to providing a framework to explain the non-thermal phenomenology of the source, this scenario predicts extended, moving X-ray emitting structures similar to those observed in PSR B1259-63.