When Mark Janus got his first paycheck working as a child support specialist for the state of Illinois in 2007, he was stunned to see a nearly $50 deduction for union dues. He had worked for the state in the 1980s and didn’t remember anything like it. “I’m going like, ‘I’m not a member of the union. What’s going on here?’” he said. Worse yet was where he saw the union dues going, such as to efforts demanding wage increases at a time when Illinois was struggling with a crippling pension debt. Now Mr. Janus is taking his case to...