WASHINGTON — Her image has been plastered on banners in Tahrir Square, crossed out with a blood-red X or distorted and smeared with insults. She is too cozy with Egypt’s deposed president and the Muslim Brotherhood, the signs say, and should leave the country. Anne W. Patterson, a press-shy career diplomat who has been American ambassador to Cairo since 2011, suddenly finds herself a target in a dangerous political upheaval, a symbol for angry young Egyptians of America’s meddlesome role in their country’s affairs. With the Egyptian military ousting President Mohamed Morsi on Wednesday, Ms. Patterson will have to navigate...