Never played the one on the Jaguar console. I have it (the officially-licensed game) for the PC although it never worked quite right on my computer for some reason.
Postal. The anti-hero walks around in a long duster shooting innocents who scream and beg for help/mercy. The victims moan, “I can’t feel my legs.” With a single keystroke, your player says, “I regret nothing,” and blows his head off with a shotgun.
The game had a “security cam” POV, like Diablo. Apparently a FPS sequel is forthcoming.
Most have mentioned games that are disturbing because they’re scary–I love scary games, but I’m generally not one to find them disturbing. Weird amorality issues, though, is disturbing. Haven’t played GTA3, but I’ve heard about it–too chilling for me. Two disturbing old-school games are Syphon Filter and Final Fantasy Tactics. I don’t know how to do the cool Spoiler box that Enigma42 did, so I’ll just say:
SPOILER WARNING!!!
and leave some blank spaces…
At one point in the game Syphon Filter, your mission is to infiltrate a laboratory, shoot any scientists you see, and inject test subjects with a vaccine. A few of the scientists shoot back at you, but most fall to their knees, cover their faces, and shout “Please don’t kill me! I’m unarmed!” Of course, you can’t finish the mission without shooting them. Also, it later turns out that the “vaccine” you use to save test subjects was actually a lethal poision: your boss was trying to stage a cover-up. My wife and I felt really icky after that level.
But nothing can beat Final Fantasy Tactics. After watching the main character’s family destroy itself after making repeated selfish, unjust, and murderous choices, and after watching the hero’s former best friend turn into a greedy despotic tyrant–you get to take your whole party to Hell, engage the game’s Christ-figure in mortal combat, destroy Him/Her/It, then everyone dies in Hell. The ex-best friend kills his wife, whom he had used to gain political influence. And the historian/narrator who was telling the game story is burned at the stake by the Church for trying to reveal the truth.
I was so upset by this ending that I skipped school for a day and didn’t eat. 
That’s disturbing!
There was an Alien patch for Duke Nukem 3D that scared the bajeezes out of me. The scariest thing was a part that was in almost complete dark and you can hear in the distance the tappity-tap of a facehugger running toward you. I never saw the patch for Quake 2, but if it was anything like the Duke Nukem patch it was terrifying.
Uh, nobody mentioned Max Payne? All about revenge on a baby killer. Very graphic.
And Enigma42, how’d you do that awesome “spoiler” thing?
Yeah, Final Fantasy Tactics has the all time ending. God, I love that game.
How to do spoiler tag:
[spoiler][/spoiler] <—like that!
Now you’re gonna wonder how I did that. Sheesh! It never ends… 
Just for the record, the Jaguar version was (a) officially licensed by Fox, and (b) is the direct predecessor of the PC version – the same software company (Rebellion) did both, and both share the same core idea of letting you play as any of the three different species.
I used to play this one on the arcade machine in our local mini-market when I was 15. Mostly it was fun - splatting green zombies with iron bars etc. But when it came to the bit with the skinless babies hanging on meathooks you had to kill with a chainsaw, I admit to feeling a little bit disturbed.