I TOTALLY thought that you were saying that you prefer a camera suitable for video gaming for your chicken monitoring and was VERY confused for longer than I care to admit 😄

100% whatever trailcam fits your budget. They are robust against weather and other insults, crazy battery life, and some have remote monitoring. Don't overthink this one.

I think this is a case where trying to actually research a product may be more trouble than it's worth, and you'd probably be better served by just walking into your local sporting goods store and grabbing the cheapest game camera they have on the shelf is.

I imagine there's not going to be a ton of ambient light available in a nest box, so you're probably going to be relying on night vision, so picture quality is probably going to be kind of shit no matter what, and since it sounds like you pretty much just need to see when a chicken goes into the box, no sense spending money for higher quality cameras.

These cameras are meant to be left outside exposed to the elements strapped to a tree or whatever, inside a nest box is more protected so IP rating and such isn't going to be much of a concern.

I think whatever you can grab off the shelf will do the job fine.