The drop on egg production could be caused by many things, such as less daylight hours, as one example.
Yes, a broody hen will stop laying eggs because she is brooding the ones she is going to hatch. Squawking or clucking is something they do while they are on the nest to "talk" to their eggs. But if you listen to your chickens, they have very different sounds for different conversations. One sound is "I just laid an egg." An other is "get out of my nest, I need to lay my egg next." Yet an other sound is "Hi, how's it going today?" Each sound has different meaning and for different reasons. So a broody hen will have yet an other sound for when she is talking to her eggs.
The reason why I am giving you all of this info is to help you understand that just because a hen lays an egg, that does not mean that she will want to hatch it. That is just one of the disadvantages to the natural method of hatching.
Some hens will sit on eggs for 2 weeks and then quit, killing all of the undeveloped eggs. Some hens will take eggs from other nests and hatch out every one's eggs. Some hens will knock eggs out of the nest accidentally and sometimes purposely. But if you let a broody hatch eggs for you, you will never have to worry about power outages or temp spikes or humidty. Things that you do need to worry about with an incubator.
I was hoping more folks on the forum would join in and offer their advice as well, but until they do join in I hope that this info is somewhat helpful to you.
