Revealing mistake: When the bus pulls up for the Taylor family to leave for Hollywood you can see behind the bus a California freeway with cars moving in both directions. I don't think that Mayberry had a freeway right outside town. The scene is fixed when they return in "Aunt Bee on TV." There has been a tree put there so the road wouldn't show.
Continuity mistake: When Warren parks the squad car at the bus stop, as he gets out of the car note the long whip antenna is attached to the bracket over the rear passenger door, but when it cuts to the closeup the whip antenna has vanished. (00:21:25)
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.