The letters keep coming, as do the emails. They head, unopened, straight into Jason Osborne’s trash and deleted folder.

The U.S. government desperately wants Mr. Osborne and his wife to start repaying their combined $46,500 in federal student debt. But they are among the more than seven million Americans in default on their loans, many of them effectively in a standoff with the government. These borrowers have gone at least a year without making a payment—ignoring hundreds of phone calls, emails, text messages and letters...