I think if it were rats that took your hens eggs, they would have eaten them, and would not have been found frozen in the woodpile.I may be wrong, but dont believe a rat would just leave the egg uneaten.
Sorry for your loss to the weasel, you'll need to secure your coop with rat wire.Weasels will get in through rather small holes if found. I've also found having a couple outdoor/barn type cats around helps with the smaller 4 legged predators..
Weasels aren't easy to trap, you may be able to live trap one with a live chicken as bait, done in a way as to protect the bait chicken.( in a seperate , very small meshed wire cage)...I've never done it, but read about someone here who has caught a weasel this way. my be in the achives somewhere....good luck...