My J305βγ tube is very light sensitive so I made a quick video demonstrating this. I shine daylight (on a miserable cloudy grey day) on the tube at first, then a 365nm UV flashlight and finish off with daylight again. The effects seem to work only at the anode end and not at the cathode end. Doing a little research I found that the light sensitivity of GM tubes only occurs in glass walled tubes that have an external cathode. The specifications show this about the tube wall: Tin oxide Cathode, Coaxial cylindrical thin shell structure (Wall density 50+-10cg/cm2). The cathode is on the exterior of the tube and you can see in the picture that is doesn't cover the whole length of the tube, but has an insulating gap at the anode end of about 7mm. The cathode is connected to the terminal cap by a little wire glued down with some kind of resin. (The resin looks like JB Weld!)
It is hard to see, and much easier by eye than camera, but you can just make out the end of the Tin oxide coating on the outside of the tube.
Here is the video I made:
For those interested, here are the specs of the tube: