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Uh.....this is not a snake M.O. by
any stretch of the imagination. Nor will rattlesnakes eat eggs. They aren't physically built for it, nor do they seek such food, and most likely ingesting an egg would kill a rattlesnake if it was unable to regurgitate. They completely lack the supporting musculature colubrids have that do allow them to crush and digest an egg. Rattlesnakes are also not big bird eaters, and are far more interested in rodents, though some specially adapted species prefer lizards. If you're seeing rattlesnakes around your coop, they want the rodents that are eating the feed. They do not want the birds and will pass up even a plump fuzzy little chick to pursue a mouse or a rat.
Rattlesnakes will not want anything to do with the adult birds whatsoever and will probably give them a wide berth. A decent sized adult rooster could very easily kill a young rattlesnake, and put a serious hurting on an older one, and they know it. Their strikes are not very effective against feathers or scaly hard leg skin. Some bird species (specifically the desert "roadrunner" types and some kinds of jungle fowl) prey heavily on snakes including venomous species, and the snakes are basically helpless against them. Feathers are a surprisingly good armor against the way rattlesnakes strike, and their legs are very nearly immune to venomous snake bite.
Colubrids are properly equipped for bird and egg eating and it's possible you have a culprit there, but there is no North American colubrid that can kill or even mildly annoy a goat. It would be the other way round, pretty much instantly, if the goat decided to pick a fight. Also, the carnage is pretty unmistakable if they kill but fail to succeed in eating an adult bird as they occasionally do. You might not recognize the marks, but you'd be describing something very different from "not a mark on them".
About two weeks ago one goat in the pen was found dead in one of my nest boxes (small goat, big boxes, so he would sleep in there at night) curled up like he'd gone to sleep and just never woke. I could not find anything obviously wrong. Earlier this week, two Silkies were dead in the coop for no obvious reason. The only snake we have here that will kill animals like that is a rattlesnake, and that seems the most likely explanation for the deaths. Two neighbors have lost animals to confirmed snakebites in the last few weeks too. So, would the rattler be eating eggs too, or am I possibly dealing with two pests here? Currently my nest boxes are on the ground, tomorrow I will move them up off the ground to try preventing snakes from getting in so easily.
This is not a snake M.O. either. Tell me exactly where you live and I'll narrow down the species in the area, but if you're in North America you are very, VERY unlikely to see animals mysteriously dead without an apparent mark on them from snakebite. There should be significant swelling and local necrosis, and it takes quite a long time for this to be fatal, especially in hoofstock which are markedly resistant.
EDIT: Apparently you're in one of the few localities where there are significantly neurotoxic vipers, so it's not impossible, just unlikely. Same story on Crotalus scutulatus not being interested in birds or eggs however. I've worked with this species pretty extensively in the wild and in captivity for venom research and antivenom production. With a "scut" bite in some localities, the ones that have the Type A venom, you might see minimal local symptoms prior to death, but swelling still occurs almost immediately even with the 35 percenters, the high end neurotoxic Type A venom producers that we want most for venom extraction. That venom is still pretty hard on tissue as it contains cytotoxic components, and you will definitely see its effects in the majority of bite cases to animals bitten in an extremity. An intracoelomic bite such as may be delivered to a much smaller animal (a rat for instance) would reliably kill without producing local symptoms, but it is not physically possible for a medium-small rattlesnake species to get an IcE delivery on something the size of a goat. They'd have a hard time doing it with something even as big as a rabbit due to the physical mechanics.
One thing you can do if you suspect your chicken was killed by a rattlesnake is to pluck it and observe the corpse for a day or two. In the fridge is fine. You should see postmortem discoloration developing due to venom in the tissues. A veterinarian should also be able to readily observe on necropsy whether there was any bolus of venom delivered into any of the animal's tissues. It does cause significant disruption in tissue.