What does the GUN button on my control panel actually do?

That button is almost certainly for a coolant gun or air blast gun, not something the old owner dreamed up. A lot of machining centers came from the factory with a manual spray gun option, basially a flexible hose off the coolant pump with a trigger valve, and the control has a button plus a dedicated output for it so the pump or solenoid only runs while you're holding the nozzle. The green and red pair underneath is usally run and stop for the coolant system itself, or in some cases for the chip auger. On my Haas the gun works thru an M-code and a foot pedal too, but on Fanuc controls it's often just that panel button feeding a relay.

Easy way to confirm without opening anything: put the machine in MDI or diagnostics and watch the input screen while you press the button. If an input bit flips when you hit it, the wiring is alive and it's a factory function. If nothing moves anywhere, either the circuit was pulled or it was never wired at your machine. If it turns out something got removed, don't just hook a new sprayer to those wires. First kill the main disconnect, lock it, and give the cabinet a few minutes before anybody goes in, drives hold charge long after shutdown. Then trace where that output wire actually lands. On mine there was a capped stub of air line near the spindle head and an empty spot on the manifold where a solenoid used to sit, which told the whole story in about ten minutes.

Honestly, worst case is somebody spliced into that circuit for a light or a fan and the button now does something random. I've seen that on used machines more than once. Trace it before you trust it, and don't feel bad about leaving it alone til then. A button that does nothing beats a button that does something you didn't expect.

I'd push back on the 'almost certainly a factory option' part. I bought a knee mill a few years back with a mystery button just like that, and it turned out the previous guy had wired it to some aftermarket mist rig he yanked when he sold. Half the used machines I've looked at have a Frankenstein cabinet behind the panel, so calling it factory from the button alone is a guess, not an answer. Also, watching the input screen only tells you the switch works, not what the output ever fed. I'd chase the wire from the terminal end first before assuming anything about how it was built.

Did the same thing you're doing now on my old VMC, just left the mystery button alone and stuck a strip of tape over it with 'DO NOT PRESS' written on sharpie. Ran that way for three years until I finally had an electrician out for other work and had him trace it in twenty minutes. Turned out it fed a solenoid for a wash gun the second owner had stripped out, wires were still live at the cabinet side. One thing worth checking before you go any further though, look around the inside of the enclosure for empty mounting holes or a blanked off bulkhead fitting. Where hardware got removed usually leaves a scar, and that'll tell you if anything was actually hung off that circuit.

Mine had a similar leftover button and I ended up just capping the wires at the cabinet with the breaker off, then forgot about it for a year. When I finally wired in a cheap trigger sprayer off the coolant line, I didn't even touch the button, just teed into the existing line. Honestly the manual hose does 90% of what I need anyway, so now that button's just decoration. If your coolant pressure's decent at the pump, that's all the gun ever was really.