I was reading the
Mesa City Code 8-6-21 and it doesn't appear they have the authority to simply tell you that you have ten days to get rid of them. The only restriction I see says:
As you have not mentioned that you were taken to court and found guilty as a "habitual offender," I don't believe they can order you to do anything. They have to tell you what the problem is and give you the opportunity to resolve the situation. Just because a neighbor "doesn't like [your] chickens" does not mean they can violate your rights defined in the City Code. If they are loud, especially the rooster, definitely get rid of him. Otherwise, as long as the coop is within the defined codes and you don't have a nasty stench or fly problem, they can kiss off.
You can read the specific laws by clicking the Mesa City Code link above. Hope this helps!!!