Oh yes, you can make a moving chicken... but that requires modelling...
The chicken is an animated model, it's idle animation is to continue bog it's head up and down. You could animate it to run around, have several different idle animations play randomly so it isn't too repeating. This is done in several maps with birds... de_plaka is the best example I remember now.
A 'problem' would be that chickens couldn't be made killable as func_breakables can't move. Also, players can walk trough the chickens. Sure, you could make an invisible train moving along the chicken, but with random animations that's not going to work properly.
So, yes, possible, but requires some skills...