Egg shells are a great addition to worm composts. They add calcium which help prevent excessive acidity. They add grit to help worms breakdown food. And I’ve read that calcium also plays a role in worm reproduction. Not sure if it’s true but I’ve never had a shortage of worms in my bin and my bin is littered with egg shells.
I never bother to break them up as some sites suggest though sometimes I’ll push them down which breaks them up slightly. The compost isn’t as uniform and with so many eggs shell bits in my bin but I tend to take a lazy approach to composting. Collecting shells separately and crushing them seems like too much effort and I don’t think the worms are complaining.
Paper egg cartons are also great for the bin. I sometimes use them as cover instead of newspaper because it gets soggy slower and allows for more air circulation than wet newspaper which sits right on top of the compost. If I do use newspaper I tend to let it sit dry on top loosely and sink down as the paper gets wet from the condensation in the bin.