Visible crew/equipment: When Barney and Andy are waiting for Virgil to arrive, just as the bus pulls up and stops, the reflection of the boom pole/mic is visible on the windshield. (00:03:50)
Revealing mistake: When Barney pulls the jail cell window out of the exterior brick wall, as it's being yanked out we can see that the brick wall is only thin drywall during the stunt, as well as the fact that the wall around the breakage differs significantly between the closeup and wide shots.
Other mistake: Barney's cousin is coming to Mayberry but gets off the bus in a previous town and misses it. Barney and Andy go out to find him, leaving from Mayberry. They are driving down the road when they come upon his cousin. The cousin is walking in the same direction as Andy and Barney are going, which would have meant he was walking away from Mayberry when he should have been walking towards it.
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.