MONSEY, N.Y. -- In his new suburban American home, Shaker Yakub, a Yemeni Jew, folded a large scarf in half, wrapped it around his head and tucked in his spiraling side curls. "This is how I passed for a Muslim," said the 59-year-old father of seven, improvising a turban that hid his black skullcap.

The ploy enabled Mr. Yakub and half a dozen members of his family to slip undetected out of their native town of Raida, Yemen, and travel to the capital 50 miles to the south. There, they met U.S. State Department officials conducting...