Daycare is very rare over here, but I would imagine certain events must be covered by the company's insurance. getting sick, unless mixed with a known tio be sick dog would not be their fault , as would breaking a leg while running. However, they would be at fault if they left broken glass lying about or left him with an unsafe toy.
I'm not sure about escaping. Our small vet opractice was taken over by a multi-sited one, some months ago, and there was an item in the evening paper reporting that they had lost a dogwho escaped while they were transfering him to their van, in order to take him to their overnight facility. In this case , I would say they were responsible, as they hadn't put him on a lead. I don't know whether the dog was found again.
Barbara, Monty, Joly and Teddy.