Home day cares even if unlicensed, are often run by early childhood educators, who run an early education program so please be mindful that the tone of your post is such that it suggests unlicensed = poor quality. It just seems that you have picked poorly on this occasion.
Legally, even with an unsigned contract, you have a care agreement in place. This is proven by the fact you attended that day home for 4 days. You would be in a stronger position, had you signed the contract since you would have a black and white legal document that proved what the parameters of the agreement were. Without it, your agreement is verbal and it comes down to a he said, she said situation.
While I agreed that the level of damage is unlikely in such a short period of time, and that the lack of evidence is troubling, I don't think your post is detailing all the pertinent information. It makes no sense, That a child attends a day home for 4 days, has zero issues because he's used to care and then out of the blue, a claim for damages is made by the carer. Something had to have happened. It's just too random to be made up entirely.
If you want to provide all the facts about what actually happened, I think people will be happy to try and help you. Alternatively, contact a lawyer.