In Brave New World, John quotes Shakespeare: "O that this too too solid flesh would melt, Thaw and resolve itself into a dew." Why?
John quotes Shakespeare many times in this excellent novel, and you would do well to consider why Huxley gives him the allusions that he does at particular moments of the novel. This allusion that you refer to of course comes from Hamlet's first soliloquy in Act I scene 2, when Claudius and Gertrude have left Hamlet alone, and he is so unhappy about the death of his father, the swift marriage of his mother to his uncle and then his disinheritance that he contemplates suicide, wanting his "flesh" to melt. John therefore uses this allusion to express his own terror and horror and unhappiness of this "brave new world" that he has been brought into and to express his own desire to kill himself.