The fly's appearance in this story is important because of the number of things that the fly and its death could be symbolic of. One thing to remember is that the fly's death was awful. The boss tortured the fly with the ink drops until it could no longer handle the abuse and died. This brutal death could be symbolic of the brutality of war that kills so many people. Keep in mind that both the boss and Woodifield lost their sons to the war. Their environment during combat was likely hellish, brutal, dirty, and painful, and that is what the boss subjects the fly to until the harshness of the environment kills it. It was only a matter of time until the fly died. Similarly, it was only a matter of time before the war claimed the lives of the men's sons.
A reader could make an argument that the fly's death is significant because it is symbolic of the wider human condition. The human condition is the totality of the human experience while living. It encompasses the highs and lows of life. It includes suffering and ultimately ends in death. The fly's death shows that it is a mortal creature. This is the same for all humans. This was shown to Woodifield and the boss with the death of their sons, and it is something that Woodifield and the boss are wrestling with more and more as they find themselves aged and nearing the end of their own lives.