There are a few things to consider when deciding whether to use nails or screws for a chicken coop. The first is the material the coop is made out of. If the coop is made out of thinner materials like plywood, then screws will likely be the better choice as they will hold the material together better. Nails can be used in this instance as well, but they may not hold as well and could come loose over time.
Another consideration is the size of the coop. If the coop is on the smaller side, then screws may be the best choice as they will provide a more secure hold. Nails can be used for smaller coops as well, but again, they may not hold as well and could come loose over time.
The last consideration is the climate.