My guess is (lucky you) they simply don't like rats. It's like the kid that doesn't like mac and cheese. Except this time the mac and cheese had arsenic. (arsenic is not really used any more)
Most modern rat poisons are anti coagulants (blood thinner / typically Warfarin). If your birds ate it, They probably die of internal hemorrhaging in large doses If they did. In small doses, you have blood thinning eggs for 2-4 weeks which could also be dangerous. The safe side would be to throw out the eggs for about a month. If you see uncontrollable bleeding from small cuts or more than normal bleeding from your chicken this may be a result of rat poison.
While the chicken eggs would probably have a low dose of it and not be therapeutic level it is a "high alert medication" in any medical text book and is very complex to take with a lot of drug interactions that can be a severe medical emergency. Ask anybody on it, they will give you an earful how much the med stinks.
However your rats nest may be near your chicken coop. They like to die near there nest, food, or water source. Mostly near their nest. They are going to their favorite places.(The bar, buffet or home.)