The Department of Housing and Urban Development proposed a new rule Friday to make people seeking housing assistance have to prove their legal status, in a move that could oust thousands of illegal immigrants. Secretary Ben Carson said the rule would bring policy into compliance with the law, which generally tries to restrict public benefits to citizens and legal residents. “There is an affordable housing crisis in this country, and we need to make certain our scarce public resources help those who are legally entitled to it,” Mr. Carson said. When the Washington Times broke the news of the rule...