If you can, talk to the neighbors. Let them know that although you really don't care if they have chickens or not, you really need them to keep the chickens on their own property and off yours. Let them know that the chickens have destroyed your flower beds and garden. AND let them know that this an expense for you as flowers and garden starts are expensive.
I envision something like this: "Hi, I am your neighbor from across the road. I see that you have chickens. I don't want to cause any problems, but we have a problem. Your chickens are spending alot of time in my yard and on my porch. It will cost me $200 to replace the flowers that they destroyed and since they have been doing damage in the veggie garden also, I am not sure I will have any home grown produce to harvest - Man, I love me some homegrown tomatoes. And, while I appreciate the free fertilizer they are leaving behind, I am not happy about slipping on it or tracking it into the house. Please figure out how to keep your chickens out of my yard." (this is close to word-for-word a conversation that I had with a neighbor once)
Give them a chance to fix the issue - like a week after you talk to them, if nothing changes, then file a complaint with your city. Call the City hall and ask who you need to contact about your neighbors chickens being a nuisance on your property. In the mean time, take pictures of the damage and birds in your yard. A photo is worth a thousand words. (Don't forget pictures of poop on your shoes)