WASHINGTON—Twenty-one federal employees who had been working with Elon Musk’s Department of Government Efficiency resigned, writing in a letter that they wouldn’t offer their expertise to overhaul the government if it meant undermining essential services.

The mass resignation of engineers, data scientists and product managers is a temporary setback for Musk and the Republican president’s tech-driven purge of the federal workforce. It comes amid a flurry of court challenges that have sought to stall, stop or unwind their efforts to fire or coerce thousands of government workers out of jobs.