I mean... Totally valid question!
IMO, it'd almost certainly be a terrible idea because all the ring has to do is get to someone else. If they lose track of that chicken for a second, I bet it'd run it's ass off and find some other creature/character that'd be easier to influence than Frodo.
Remember: It is not the petite, easily overpowered nature of Hobbits that made them a good choice. It was their peaceful, carefree nature: They are WAY less temptable by the idea of power and control than others.
So... While the worst damage a chicken would be likely to do directly would be to peck out the eyes of Sam and Frodo while they slept, it would still be foolish to put the ring in control of anything more temptable or less driven to destroy the ring than Frodo.
