We see the linkage to the hivemind severed in a number of different episodes, notably in the TNG episode "I Borg" and the Voyager episode "Collective"
In "I, Borg" a single drone is removed from the collective and his linkage broken using a subspace field. After a brief period of confusion his personality begins to reassert itself to the point that he is ultimately willing to sacrifice his life to allow his newfound friends to escape.
In "Collective" we see a group of borgified children removed from the Collective. Despite the efforts of their leader ('First') to establish order and reconnect to the Collective, they ultimately become individuals again, remembering their names and individual personalities.
It's fair to assume that any weapon capable of severing an individual cube's connection to the hivemind would be largely ineffective in the short term. It appears to take several days for the borg to start to become individuals again, and even longer when there are more than one.
Unless any hivemind blocking technology was backed up with further action to prevent the cube from reestablishing the link (and thus providing the occupants with sufficient time to begin the process of regaining their individuality) they would simply repair their cube, adapt to the blocking technology, reconnect to the collective and continue with their original mission.