Gregg Schwarz frowned as he positioned himself, just so, in front of the wrought iron fence surrounding John Edgar Hoover’s grave, a place he has visited countless times but never before in anger.

A retired FBI agent who joined the agency in 1972, the year Hoover died, Schwarz had hired a videographer to film him for YouTube expressing his displeasure with a movie that depicted Hoover as a repressed homosexual. In a dig at Clint Eastwood, the director of “J. Edgar,” Schwarz titled his video response, “Dirty Harry to Filthy Harry.”

“Mr. Hoover was portrayed as an individual who had homosexual tendencies and was a tyrannical monster,” Schwarz said into the camera, as the sun glinted off his FBI cuff links and FBI lapel pin. “That is simply not true.”

Many former FBI agents share Schwartz’s pique with the film’s dropped hints of an abiding love between Hoover and aide Clyde Tolson, who is buried a few grave sites away. Historians agree that there is no evidence that either man was gay, and a request for comment from either Eastwood or screenwriter Dustin Lance Black was declined....