I have to give credit to this article, which phrases it better than I could.
Dwayne is Olive’s angsty teenage brother, shown early on reading Nietzsche’s Thus Spoke Zarathustra and wearing a shirt that reads “Jesus Was Wrong.” He even has a wall-sized painting of Nietzsche hanging on his bedroom wall. Dwayne takes a vow of silence inspired by the great philosopher, using Nietzsche’s radical energy and idea of the Übermensch to channel his disdain toward others. As an angry teenager, Dwayne understands nihilism as a reason to hate the world and everyone in it.
In short: Dwayne hates the world (in classic teenager fashion), and expresses that by partially refusing to engage the world, essentially giving it the silent treatment.
But at this stage Dwayne misunderstands Nietzsche, whose nihilism was not a rejection of life but an affirmation of an authentic life unbound by false values.
In short, Nietzsche's actual conclusion is closer to the ending of the movie, where the family all have decided to let go of their expectation and instead live life as it comes.
Coincidentally, Nietzsche spoke out against keeping silent, a point that very much highlights how Dwayne's silence is not in line with Nietzsche's thinking:
- "Silence is worse; all truths that are kept silent become poisonous."
- "For both parties in a controversy, the most disagreeable way of retaliating is to be vexed and silent; for the aggressor usually regards the silence as a sign of contempt"
To be fair, Dwayne does actually feel contempt for the world (in classic teenager fashion), but he seems to have cherrypicked Nietzsche's words by choosing his vow of silence.