PHOENIX — As the sun rose on another day of record-breaking heat, Juan Gutierrez and his construction crew were already sweating through their long-sleeve shirts. It was 91 degrees, and workers in a subdivision called Desert Oasis were racing to nail together the wooden skeletons of $380,000 homes that had sold before they were even built.

“Your skin falls off, you have to cover up everything,” said Mr. Gutierrez, 22, who has been undocumented since he came to the United States as a 4-year-old. “It’s work you have to do. You have no choice.”