A hen cannot count, so she might not sit until there are 30 eggs or she will sit on nothing at all. Going broody isn't about her seeing eggs in the nest, it is entirely dictated by hormones. You can't make a hen go broody, it only happens if she is hormonally triggered to do so. It is also a breed thing - in some breeds like leghorns broodiness has been almost completely bred out whereas others like silkies are extremely likely to brood. If you're trying to save eggs for her, don't wash them. This takes the bloom off and can allow bacteria into the egg, causing embryo death. Storing them in an egg carton is okay. The best conditions for egg storage for hatching are at 59 degrees Fahrenheit and turned once a day. However, after ten days fertility starts to decrease and the chances of the egg hatching is reduced. If you just want a hen to sit on them though leaving them in the nest is fine. Just mark what date they were laid on them and once you get the amount you want her to sit on just take out the oldest ones as new ones are laid.
But yes, you can leave the eggs in as long as you are marking them and removing the old ones. A nest with eggs in it is more appealing to a bird that is going broody than an empty nest, but it won't trigger a hen to brood.
