And after, when
We went outside to look at her finished lantern
from the road, I said I liked the way her light
shone through the face that flickered in the dark.
—"Jack O'Lantern," Katrina Vandenberg in Atlas
People say friends don't destroy one another
What do they know about friends?
—"Game Shows Touch Our Lives," The Mountain Goats
We have two epigraphs serving as a road sign welcoming us to Paper Towns (population: ?).
The first is from the end of a poem in Katrina Vandenberg's collection, Atlas. In the poem, it seems that two people have carved pumpkins and are admiring their handiwork from the road. In Paper Towns, there are no pumpkins—it isn't even Halloween. However, we have the image of a face flickering in the dark, which happens twice in Paper Towns. In the prologue, Margo Roth Spiegelman looks at Quentin through his window, and the novel's last line says how Quentin "can see [Margo] almost perfectly in this cracked darkness" (3.22.178).
So, is Margo Roth Spiegelman a pumpkin (a Margo Jack o'Spiegelman)? She is kind of a man-made construct. No one really knows who she is, and her identity, even to herself, is composed of what other people think of her. So maybe the epigraph is about seeing through this and glimpsing her inner light.
The second epigraph is from a song by Quentin's (and John Green's) favorite band, the Mountain Goats (whose lead singer's novel was nominated for the National Book Award in 2014). Quentin and pals listen to the Mountain Goats (maybe even this very song) in Part 2, Chapter 8, on their first early-morning trip to Margo Roth Spiegelman's mini-mall (the Margo Roth Spiegelmall, home of the Margo Roth Spiegelmall holiday catalog).
The quote in the epigraph is misleading, though. Quentin and his friends never destroy each other, and they only get into, like, one fight that lasts about six paragraphs. Margo Roth Spiegelman, however, tries to destroy people, but you could argue that she never had any friends to begin with, since she's hollow inside like a carved pumpkin. Perhaps what gets destroyed is her ability to live beneath a façade—perhaps, somehow, her friends (or lack thereof) force her to quit being fake and invest in being real.
So we have two epigraphs only loosely related to the actual content of the book. Basically, they're both epigraphs by people John Green admires, and he wants you to read and listen to them, too. So get on it.