Preserving Border Integrity Act This bill imposes restrictions on non-U.S. nationals (aliens under federal law) seeking asylum in the United States. Under this bill, a non-U.S. national may not, except in certain situations, seek asylum if the individual (1) entered (or attempted to enter) the United States through the northern or southern border, and (2) did not apply for protection from persecution or torture in at least one country that the individual transited through to reach the United States.