What role does national identity play in The Crossing, specifically in Billy's struggles with national identity towards the end of the novel as well as divisions between America and Mexico that become apparent at the beginning of the novel?

In the novel, conflicting national identities highlight divergent approaches to life. Basically, the role of national identity in The Crossing is to draw attention to the stark differences in philosophy that separate two cultures. The rugged individualism of the American West is personified in Billy Parham (our sixteen year old protagonist), while the primitive wildness of the Mexican landscape is embodied by the she-wolf.
At the beginning of the novel, Billy manages to track down and catch a pregnant she-wolf. He tells us that the wolf had crossed the "international boundary line" to track her own kind: "She was moving out of the country not because the game was gone but because the wolves were and she needed them." Billy asserts that the she-wolf knew nothing of "boundaries."
Her main purpose was to reconcile with her own kind and to thrive. On her arduous journey to New Mexico, she subsisted on nothing but carrion for two weeks. By the time she reached the Peloncillo Mountains, she was close to starving. So, her feasting on veal calves was a desperate attempt to survive, however cruel the devastation caused to the American cattle ranchers.
For his part, Billy willingly braves the perilous journey to Mexico to return the she-wolf to what he considers her natural habitat. At a local fair in Mexico, Billy gets his first inkling that he has deeply underestimated the political divide between America and Mexico. A local hacendado (farmer) quips "You think that this country is some country you can come here and do what you like." The question of differing national identities and psyches exerts itself: Mexico is a separate country with unique social codes, and borderlines are not just geographical boundaries; they also mark the divisions between cultures.
Billy learns this too late: his beloved she-wolf is soon confiscated by Mexican authorities and turned over to fight packs of dogs for sport. By the time he manages to get to her, "her head lay in the dirt and her tongue lolled in the dirt and her fur was matted with dirt and blood and the yellow eyes looked at nothing at all. She had been fighting for almost two hours and she had fought in casts of two the better part of all the dogs brought to the feria." Incensed at the cruelty of the Mexicans, Billy takes up his shotgun and performs a mercy killing.
By the end of the novel, Billy has crossed into Mexico three times. His first journey was made to return the she-wolf to Mexico, the second journey was made to recover the family's stolen horses, and the third journey was made to retrieve Boyd's body. Each of the journeys reinforces in Billy the lesson that borderlines and national identities have very little to do with personal identities. In his conversation with Quijada towards the end of the novel, Billy comes to understand this.
When he tells Quijada that he wants to take Boyd's body back to America to "bury him in his own country," Quijada tells Billy that "the dead have no nationality."

And it is because these names and these coordinates are our own naming that they cannot save us. That they cannot find for us the way again. Your brother is in that place which the world has chosen for him. He is where he is supposed to be. And yet the place he has found is also of his own choosing. That is a piece of luck not to be despised.

So, in the novel, the subject of national identity has two main roles:
1) It highlights the political and social divide between two cultures.
2) It stresses that boundaries cannot circumscribe (limit) the human spirit. 
Source: The Cambridge Companion to Cormac McCarthy edited by Stephen Frye.