HONG KONG—Japanese citizen Midori Nishida was checking in to a flight in Hong Kong in November to visit her parents on Saipan, a U.S. island in the Pacific, when airline staff made an unusual demand. She had to take a pregnancy test if she wanted to board.

Ms. Nishida, 25 years old, was escorted to a public rest room and handed a strip to urinate on.

The...