After Duke Tran escaped from slavery, but before he became a millionaire, he was a Wells Fargo employee.

He worked at the bank’s debt-collections center near Portland, Ore., talking on the phone to customers who owed Wells Fargo money. It wasn’t glamorous, but the job enabled him to afford a two-story suburban house with mustard-colored aluminum siding. After more than three decades in the United States, Mr. Tran felt that he was the living embodiment of the American dream.

And then it all started to crumble.

In 2014, according to Mr. Tran, his boss ordered him to lie to customers who were facing foreclosure. When Mr. Tran refused, he said, he was fired. He worried that he wouldn’t be able to make his monthly mortgage payments and that he was about to become homeless.

Joining a cadre of former employees claiming they were mistreated for speaking out about problems at…