Cattle, horses and sheep do not have a central parting in the hair on their back, goats do have a central parting like us. Therefore when it rains, if they are out in it, the rain can soak straight down to the skin, which if they then catch a chill can cause pneumonia. Goats do not suffer from diseases like sheep if they are housed for a long period of time. Some goats are even keep as yarded animals. Dairy cows nowadays are often kept indoors- goats can be more similar dairy cows than sheep.
Its all very well letting the goats choose where they want to shelter, but there should be a building there to shelter in, even if it is just an open field shelter. I would also say there ought to be someplace where they could stand out of mud, so a dry floor or paving slabs down would create that. I would have reported your neighbour by now (well actually I'd have told them in no uncertain terms, and if nothing changed I'd be reporting them).
Beth