Is it permissible for workers to take damaged cardboard boxes, destined for the trash or taken by workers without the management's knowledge, to their homes for packaging?

It is not permissible to dispose of food or take it without the administration's knowledge, except with their permission. Damaged cartons do not justify discarding food. Trustworthiness must be maintained, and nothing should be discarded before consulting the administration. If the administration instructs that it be discarded, then there is no harm in benefiting from it.