To box a Cylon means to permanently download a Cylon agent's consciousness from their body into cold storage (apparently a form of computer data dump).
To an agent, boxing is apparently a form of permanent death or state of unconsciousness. As Cylon agents appear to work with a collective mind, boxing is done with a Cylon agent whose memories appear "corrupted" by extended or intense human contact. The recovered Number Six (who died protecting Gaius Baltar when his house is destroyed in the Miniseries) and Lieutenant Sharon "Boomer" Valerii are threatened with boxing by a copy of D'anna Biers shortly after their return, as their "hero" status and extended human associations threaten the Cylon status quo (Downloaded).