If you have adequate weather/predator protection in the run, that's my preferred location for a breaker for the duration of the breaking period. Coop otherwise.
If placing it in the run but weather/predators are a concern and the coop simply doesn't have room, you can move the crate into the...
If she stays on the nest overnight, or begins showing behaviors such as: a tik-tik-tiking sound, puffing up like a turkey, flattening like a pancake and growling, ripping feathers from her chest, then those are signs that you have a broody and you'll need to decide how to want to proceed with that.