PThomann: In your case, you mentioned red and black wires. The red would go to positive, the black to negative. With power going into the camera, the wires have continuity (no breaks). Some electrical component must have failed in the camera to do with the external power supply. Odd that all three cameras failed but like a lot of things, maybe that part has reached its life expectantcy. Sad when one needs these cameras as hunting season approaches.
If you haven't already done so, request approximate quote from the manufacture for a cost to check out and/or repair the camera. You would have to pay shipping one way or posssibly both. You would also require a return authorization number.
If the cameras are off warranty, you could: 1)shut the switch off, 2)remove the external power supply, 3)remove the internal batteries, 4)carefully open up the camera and check for a loose or poor connection on the power supply plug socket in the camera. Also look for a discoloration or burnt smell which could indicate a failed component in that circuit.
Depending on the repair cost and shipping x 3 (cameras), it may be cheaper to put that money toward a new camera. Good luck.