Across the U.S., more employers are handing out activity trackers and rolling out high-tech wellness programs that aim to keep closer tabs on workers’ exercise, sleep and nutrition, and ultimately cut ballooning health-care costs.

Disney, Whole Foods and dozens of other companies have introduced programs to reward employees for meeting certain criteria on health indicators such as weight-to-height ratio and blood pressure. Some incentivize workers to hit a target number of step counts and eat well: One wellness provider,...