You can also use piggy bait (available at animal goods vendors) to capture it and move it to another street (just make sure to watch the timer by the map so piggy doesn't escape)! =)
EDIT: ROFL!!! So sorry!!! I am at work and should have paid more attention before posting!!! I apolgize again and good luck with your chicken!!! =^.^=
If a chicken is sad, you can help by removing any animal from the street - so you can use bait to capture a pig and that may restore the street's health, and with it the happiness of the chicken.
I prefer to use animal potion and then donate the chick. enough people flooding with world with chickens without us also finding new homes for those ungrateful chickens who don't realise how effin good they've got it.
I don't want to spoil the fun of the game too much, but do want to remind you that there are no actual animals involved. It's all just little bits of javascript.