I am not the most experianced hen keeper but a few thoughts...

First establish if it is the rats that are causing the problems. The reduction in production could mean they are unhappy (which they maybe if a rat is about)

The broken shell thing could be rats but it is also not unknown for chooks to break/eat they're own eggs. It maybe difficult to establish the culprit but is their any power struggle you have noticed as this sort of upheaval can cause problems.

Finally they can sometimes stop laying while they moult.

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