Replacing a doorbell transformer attached to power extension bar
Do not replace the junction box.  It is a standard junction box and there is nothing wrong with it.  

Backstab connections fail more often than transformers

Look at the receptacle.  See how the wires are connected?  They're "back wired" with a backstab style connection.  That is only allowed for 14 AWG solid wire, the receptacle's labeling plainly says


  PUSH IN #14 CU SOLID WIRE FOR 15A BRANCH CIRCUIT ONLY.


CU = copper.  It is then repeated in French. 

Looking at the transformer's ground wire it definitely stranded wire that has been tinned with solder to keep it from getting a case of the frizzies. The ground wire has been horked onto the already-occupied ground screw in an impressive display of bad workmanship. 

Equally bad workmanship is obvious in the attaching of the transformer wires to backstabs.  The wires are stranded and certainly not 14 AWG.  This is a violation of NEC 110.3 which requires the instructions and labeling of the receptacle be followed (14 AWG solid only).  

Since backstabs have a spectacular failure rate (open circuit) even when they are used to spec, we should expect this to be a failed backstab. 

So, I would remove the transformer wires from the backstab (if it's possible to do this while also facepalming), and move each wire to the appropriate screw terminal.    I would also pause to inspect the wire for any arcing or galling, because that would explain the mystery. 

As far as actually removing the wires, there should be a little slot into which you can jab a tiny screwdriver, and sort of pry away from the wire and that should reduce the extraction force somewhat.  Otherwise feel free to just use an excessive amount of force. Worst case the wire will break off inside the backstab and you'll have to strip off another 1/4" to attach it to the screw. 

When removing #14 wires, just twist the receptacle back and forth about 60 degrees repeatedly while using serious force.  Don't cut; wire length is precious in most cases. 

While you're at it, fix the ground

The ground installation was acceptable until this transformer came along. Now it is one too many wires on that ground screw. Move the "U" loop on the receptacle screw so you have about 2" of extra length sticking out (instead of the 1/2" now sticking out). Wire-nut the transformer ground wire to that.