It really isn't just Mrs. Dubose's comments that causes Jem to destroy the bushes. For weeks, he and Scout have been listening the disparaging remarks of all the townspeople. Jem has been encouraging Scout to turn a deaf ear and not be offended by the talk of ignorant people. She is the straw that broke the camel's back. Really, his attack was not so much an attack on Mrs. Dubose but an attack on the whole town.
Mrs. Dubose is old and infirm, and has long been considered by the children to be "the meanest old woman who ever lived" (Chapter 4). She regularly shouts abuse at Jem and Scout when they pass by her house, criticizing their appearance and speech, lamenting the death of the children's mother, and attacking the way Atticus is raising them in her place. Atticus counsels Jem especially to make allowances for the old lady's illness and not to react to her behavior, but when Mrs. Dubose vilifies Atticus for defending Tom Robinson, a Negro, screaming, "Your father's no better than the niggers and trash he works for" (Chapter 11), Jem can contain himself no longer. He destroys the old lady's camelias in retaliation for her meanness.