Look for tiny bits of egg shells and the same slime stuff in the nest and whatever you are using in the next box for a surface, like hay or pine shavings. They might be hard to spot, so look really well. I'll bet that what you are getting is egg white from a broken egg on the outside of eggs laid. This can be a couple of things - you have an egg eater who gets in there, breaks open an egg and makes the mess on the other eggs in the box while enjoying her treat; you have one overzealous layer who breaks eggs in her enthusiasm to lay (my Brahma was notorious for smashing eggs under that behemoth body of hers while she settled in to lay); or you may have one laying soft shelled eggs that can't take the weight of the next hen to come into the nest and it bursts.
Before I would start digging real deep into some exotic or serious issue, I'd always look for the simple and obvious first. If indeed that's egg on the outside of the shells, you can wash the egg with plain water, dry it well, refrigerate it and use it as if nothing had happened.
