House Speaker John Boehner squashed growing expectations that Congress would rewrite immigration laws this year—dealing a setback to the White House, a swath of U.S. businesses and a bloc of fellow Republicans seeking to improve their party's chances with Hispanic voters.

Rank-and-file Republicans have voiced few complaints about the GOP's latest ideas for an immigration overhaul, which include some legal status for most of the 11.5 million people in the U.S. illegally. But many balked at debating an issue that divides Republicans...