In October 2014, Nathan Kay applied for a job as a pilot at L3 Technologies, a Pentagon contractor that helps conduct surveillance and reconnaissance missions around the world. Mr. Kay had spent more than 10 years in the Washington Air National Guard, and had flown similar missions in Iraq and Afghanistan. A friend at L3, then known as L-3 Communications, indicated that a chief pilot was interested in his résumé. But when months went by without even a rejection — and after consulting with his friend — Mr. Kay concluded that L3 was discriminating against him because he was an active guard member.