I'm here because I throw eggshells in with my neos, but I really don't know. I don't take them out on purpose. They just sort of turn into part of my substrate/tank bottom litter.
As for adding calcium to the water, I suspect that they don't very much, only because my eggshells aren't really breaking down or are doing so very slowly. And I know that when adding them to tomatoes beds, e.g., the calcium, phosphorus, etc., is basically for next year's plants, not this year's.
Maybe if you were grinding them up, like I've seen other people recommend? I'm too lazy for that.