LONDON—British Prime Minister Boris Johnson laid out a hard-line negotiating stance with the European Union, setting the stage for fraught Brexit talks before the U.K.’s scheduled departure from the bloc on Oct. 31.

In a combative first speech to Parliament, Mr. Johnson reiterated Thursday that the plan in place to avoid a border on the island of Ireland after Brexit—one of the central planks of the divorce deal—was “unacceptable” and would have to be abolished.

That...