Witness this army of such mass and charge,
Led by a delicate and tender prince,
Whose spirit, with divine ambition puffed,
Makes mouths at the invisible event,
Exposing what is mortal and unsure
To All that fortune, death, and danger dare,
Even for an eggsheel. (4.4.47)
John's allusion is from the passage in which Hamlet observes the noble Prince of Norway, Fortinbras, who has the courage to go into a battle that he may easily lose. Inspired by this fortitude, Hamlet finally decides to avenge the death of his father by dueling Laertes and trying, then, to kill Claudius. He declares that he is "Hamlet, the Dane" and sallies forth. After alluding to this passage, John asks Mond, "Isn't there something in living dangerously?" in repudiating the new world of soma, conformity, conditioning--in abolishing "the slings and arrows of fortune."
The discourse between Mustapha Mond and John in which John argues for choice and God illustrates the dichotomy of the New World against the old world of the reservation. Here Huxley poses a choice between freedom and comfort. Fortinbras in the passage from Hamlet is noble, ready to sacrifice in the name of freedom; John chooses freedom and God. Mond chooses the stability and comfort of the New World. The two world views of John and Mond are obviously incompatible.
In this line from Chapter 17, John is quoting from Hamlet. What John means by this line can be seen a little farther down in the same paragraph. At the end of the paragraph, he asks Mond "Isn't there something in living dangerously?" In other words, he thinks that people in this society need more risk in their lives.
To John, life in the brave new world is not very human. It lacks danger, it lacks emotion, it lacks tears. He thinks that the society harms itself when it tries to prevent people from being sad or being hurt. He thinks that people need to risk things -- they need to risk being sad, for example, so that they can truly have a chance to be happy.
This quote sort of sums up one John's problems with this society -- he thinks it has given up its humanity in exchange for security.