Absolutely you may compost eggs/eggshells. Yes they do take longer to break down. Just like a branch would. If you want you could crush it up with a rock or step on it, the smaller pieces will compost faster.

Yes indeed, eggshells are good in compost. For one, they add calcium to the pile. I have read articles on growing tomatoes that actually recommend putting a crushed egg shell in the hole under the seedlings at planting time. I keep a bucket in the kitchen to collect scraps, and I crush the shells with my hands as they are added. I can't say how long they take to entirely break down, but the small particles have to assist drainage in the soil if nothing else.