The chicken thing is a one solution sidequest: you either kill the chicken, or you leave it alone. It's a bit of a test of character. If you kill it, should you see a shrink? If you don't kill it, are you too friendly to even survive in the real world? Questions, questions!
That is what I did, though not specifically to get the achievement (didn't even know about it, so that make it kind of funny when I got it). I saved before trying to herd the chicken towards Casper, but it was very uncooperative, so I killed it (knowing I could just reload). I went back to Casper thinking I might at least tell him where the chicken was, but without that option there didn't seem like there was anything left to do.
