I'm not advocating using poison but I did eventually when I had a rat infestation in one of my coops. I know that's not the same as having mice. I'm not sure why you have mice but I don't believe it has to be because of chickens. As you said you don't have any birds but you neighbor does. Somewhere they found a nice cozy place to live. If you have other pets and if the feed is left out, it can attract the varmints. I tried most everything when I first discovered rats to no avail. When I renovated my coop taking out the ceiling and walls, dozens of rats of all sizes poured poured out. I discovered several nests in the process. I bought some rat bait stations. The rats didn't care for the bait that came with the bait stations so I bought one that the feed store up the road used and the rats seemed to love that bait. The bait stations I use have a little window above the bait so I can check it often. I never found any dead rats laying around but they did have tunnels around the coops. I assume they went into their tunnels and died. As far as secondary poisoning, most pets that have consumed a mouse or rat may get sick but it's not usually fatal. Eating the poison directly may be fatal. There are responsible ways to keep pets and other critters away from the poison. I put the bait stations in a pet carrier so the only things that can get to the bait is the mice or rats. There is a patrician down the middle of the bait station that the rats have to go around to get to the bait. I keep the bait stations loaded with bait and check them often. I haven't see any rats in a very long time and the baits haven't been touched. Whatever you decide to do, good luck...