NEW YORK—Hillary Clinton met with a father living in the U.S. illegally, his wife and children on Monday, and reiterated her promise to push for a path to citizenship for undocumented immigrants.

Speaking at an immigration conference, Mrs. Clinton, the Democratic front-runner for president, practically shouted her vow to use executive power to protect people from deportation as well as to push for Congress to act on permanent legislation.

“We...