AUBURN, N.H. — They walked for miles through the woods, the girls in bonnets and long skirts, the boys in suspenders and broad-brimmed hats, towing heavy wooden handcarts. They slept in a field under tarps in a torrential downpour, and in the morning, they ate oatmeal in sodden clothes among the mosquitoes. “It was awesome,” said 17-year-old Ida Mweze of Worcester, smiling serenely. “We were singing songs, and it kept us going.”