'Corner' has meant 'share' since the middle of the 19th century. Thus, 'to stand one's corner' means to do one's share of whatever task is in hand. Of course, one's 'corner' in boxing is the part of the ring a fighter retires to between rounds.
'Boxing one's corner', I'd say, just means much the same as 'standing one's corner'. So, a newcomer to a work-place, for example, has to learn what is expected of him and ensure he does it satisfactorily, just as a boxer has to learn how to grasp the layout of the boxing-ring and act appropriately wherever he is in it.