V.B.
Here's the thing.
He sounds like he is willing with you and at home in a familiar environment, but not so much with someone else or a strange environment (he might be familiar with the daycare itself, but less so with its bathroom, since he's not really been using it, but having diapers changed on him--and it isn't likely that was happening in the bathroom).
He isn't too young to potty train, but perhaps with these parameters, it is best to put him in pull ups. My son was training at that age (and did very well), but I was a SAHM and was there to help him, because at that age, he does probably need help with his clothing and such. Not that he can't pull his clothes off, but doing it "in time" can be a big issue.
It isn't just toddlers and new potty users that sometimes have "strange potty" shyness. I have known adults that won't poop anywhere but their own home. It can really gum up the works when you are training them and they don't have access to their familiar bathroom, though. :/
Alternately, some kids LOVE trying new bathrooms. My daughter was like that.... until she happened across an auto-flush and it scared her.