We had an old short-haired mutt dog when I was a little kid. Her name was Queenie. She was right there the day my dad and brothers and other relatives came out to dig out the old heap where the old chicken coop stood on the property. It was a rat city. Queenie killed 99 rats before the day was through, one right after the other. Gross, I know, but what an awesome doggie she was. Trust me, our family was SO happy to be rid of that whole mess. Poor girl got cancer real bad and we had to put her down. So, a mutt could do it. However, terriers are really bred for that, however, also are notorious chicken killers, like someone already mentioned. So, you would need lots of training.
I personally love the Soft-Coated Wheaten Terriers and they don't shed.