Press G to equip gun.

Left click to shoot.

Okay so, this is more of a tips and tricks section that actually goes into more detail on how you can actually find greg. Greg can only spawn in areas you have already loaded, so just loading a few chunks and waiting for a little bit should be able to spawn one in quickly for you.
There seems to be a bit of confusion as to how to actually equip the gun, in order to use the gun, you must be at least be logged in to the game. The paint gun is NOT a Deluxe feature. If you are logged in to the game and still cannot use the paint gun, check to see if your keybinds are changed in the menu. The default button should be 'G'.

There also seems to be other gregs. Tim, the speedy yellow one, and the sneaky un-named green one, and possibly a even speedier red one. There may also be more gregs that have been left undocumented. These are enigmas, and are not to be trusted. Blue Greg is the only true watcher.