I'd go with Melville's "Bartleby the Scrivener." The story is so wonderfully complex that it has many meanings, but a major focus is the loss of self in the wake of the industrial revolution/urban sprawl.
Kate Chopin's The Awakening is a great fit for that theme. Her struggle to define herself separate from the roles she has play in her society bring to light multifaceted and essential questions for many people.
And I'll add on Ayn Rand's The Fountainhead. The protagonist of the novel, Roark, prides himself on his individual talent and ability to "break the mold." It's interesting to see what he both gains and loses in the process.
The little known John Steinbeck novel To a God Unknown might serve your purposes fairly well. It is a novel that examines the nature of belief, but in it we several characters who rebel from society and from family in a variety of ways. The novel is rather dark, but is nevertheless a great read.
A delightful novella of an individual who is rather eccentric, perhaps, is Truman Capote's The Grass Harp. Hemingway's The Old Man and the Sea, A Farewell to Arms, and Toni Morrison's Song of Solomon have characters who retain or find their individual souls.
You've gotten a lot of novel recommendations here, all great; I will add some other types of literature here. "discussion" mentioned Emerson, Thoreau, and Whitman, and for good reason -- Emerson and Thoreau were basically the fathers of Transcendentalism, a literary movement highly based around independent thought, and some critics also consider Whitman to be a sort of Transcendentalist. Specifically, look at Whitman's "Song of Myself," one portion of which declares the great quote, "I sound my barbaric yawp over the roofs of the world!" Thoreau wrote an excellent essay titled "Civil Disobedience", or "Resistance to Civil Government", on which Dr. Martin Luther King, Jr. based many of his writings, including King's great "Letter from Birmingham Jail". Finally, take a look at some of the great Harlem Renaissance writers, including Langston Hughes, who wrote poems such as "I, Too, Sing America" and "Let America Be America Again" (coincidentally, good partner poems with Whitman's "I Hear America Singing").
I would also try Lorraine Hansberry A Raisin in the Sun, because Beneatha's and Walter's characters struggle with trying to be their own person and not assimilating into someone else's culture or someone else's opinion on how they should be/act.
You've gotten a lot of great suggestions here already. American literature isn't my strenght, but I will add to the list: Edith Wharton's Ethan Frome and her short story, "Roman Fever". Richard Wright's Native Son might also be a good one to consider. Nathaniel Hawthorne's Scarlet Letter is another good example of this theme.
Modern books might include Keasey's One Flew over the Cuckoo's Nest; Updike's Rabbit, Run; Franzen's Corrections. These explore essentially American struggles for identity and independence in the midst of conformity and facelessness. More feminist novels might include Chopin's The Awakening and Hurston's Their Eyes Were Watching God.
Perhaps the greatest novel about American individualism is Mark Twain's The Adventures of Huckleberry Finn, which T. S. Eliot and Ernest Hemingway both agree is the great American novel. Hemingway said, in fact, it is the novel from which "all modern literature comes..." This picaresque (novel about a rogue) shows Huck on an episodic adventure, (mainly) alone in an unjust American society. Through ironic wit, Twain shows why it's better live outside pro-slavery society.
Hester Prynne (Scarlet Letter) as adulteress, forced to live outside Puritanical society
Other good choices:
The Declaration of Independence
Ralph Waldo Emerson, Henry David Thoreau, Walt Whitman