Continuity mistake: While Aunt Bee and the party guests are gathered in the parlor, when Gomer hoots like an owl in the shot facing him note Floyd's eyeglasses - black with thick straight temples, but in the shot from behind Gomer and all following shots Floyd's wearing different eyeglasses - lighter color with curved tips.
Visible crew/equipment: Just as the first shot opens of Andy walking down the block we can see the moving reflection of a crew member wearing a short sleeve, button down shirt, who's presumably sitting on the camera dolly as it rolls along keeping pace with Andy.
Other mistake: When Barney, Thelma Lou, Helen, and Andy are in the squad car talking about getting dinner, there's black tape on the chrome trim of the triangular vent window - presumably to prevent the camera/crew's reflections; and the vent window is shut in wide shots, but open in the closeup - also to prevent reflections.
Continuity mistake: While Barney and Thelma Lou are stalling Andy and Helen in the squad car, note the long whip antenna is missing (the empty bracket that always holds the antenna in place is over the rear passenger door) when they drive off to Andy's, but when they pull up to the house the antenna's on properly.
Visible crew/equipment: At the courthouse, while Barney and Thelma Lou are trying to stall for time, in the shots facing Andy and Helen when Andy says, "Steaks, chops, macaroni surprise," we can see the shadow of the boom mic moving around at the top, left side of the screen.
Answer: Those "boxes" are valences that used to be very common, before air conditioning. They allow for windows to be open during rain storms. They permit air circulation, without letting the rain in.