This style of codeword/phrase system was used in Europe during WW2 to send instructions to operatives behind enemy lines. The Chair is Against the Wall could mean that a set of railway tracks must be blown up or a supply drop was en route to a group of operatives.
"John has a long mustache" is a direct reference to the Allied code phrase for Operation Overlord, the Normandy Invasion (D-Day, WW2). The signal, like many others, was sent over normal radio broadcasts; each codeword/phrase meant something to a specific group or individual listening in Nazi occupied territory. This specific code phrase was the signal to the French Resistance (the Maquis) that the invasion would begin the next day. To them, this meant that need to execute pre-arranged sabotage and other preparations as planned before the landings on the French coast.
The whole sequence is an homage to the movie "The Longest Day", in which French Resistance members are listening to the radio, and hear a sequence of the normally meaningless (to them) phrases, until they hear "John has a long mustache" repeated twice, and set in motion the pre-invasion plans. The Germans also realize the invasion is about to begin, but due to Allied deceptions, the Nazis are still unsure as to the landing site.
In "The Longest Day", the sequence was "Molasses tomorrow will bring forth cognac", then "John has a long mustache".
