I wonder if he is sensitive to loud noises. The toilets in public rest rooms (including preschools and day care centers) tend to be much louder than your toilet at home probably is (especially if your toilet is the kind with a tank).
I have some sensitivity to loud noises even now in my 40s (I was born prematurely and had some complications from that and in addition like some other people in my family I am (at least slightly) on the autism spectrum (basically what they used to call Asperger’s Syndrome)). Your son may or may not have something like this (only a doctor or other specialist trained in this area would know for sure). On the other hand, if he spends most of his time at home he may just not be used to loud noises (like the flushing of public toilets) yet.
In any case, it would be a good idea to get to the bottom of this as soon as possible. If the issue is sensitivity to loud noises you may want to ask your son’s doctor if noise canceling headphones or ear plugs would be a good idea for him (I don’t know if children this young can use either of these options safely).
As for using regular diapers there is probably no harm in this (at least in the short term). The day care center is right about pull-ups not working as well as regular diapers for three reasons (that I know of):
1. Pull-ups are less absorbent than regular diapers that are taped on. They often contain less super absorbent polymer than the tape (or velcro) on products.
2. Pull-ups often don’t fit as well as the products that tape on. This tends to lead to gaps and leaks.
3. Young kids (and even even some older kids) will sometimes forget to pull their undearwear or pull-ups all the way up and this will inevitably cause gaps and possibly leaks. This often happens after kids use the restroom if they are in a hurry to get back to snack time, playing, etc.
I wouldn’t worry too much about your son being embarrassed about going back to diapers. It is actually still pretty common for three-year-olds these days to still be in diapers. I would still get to the bottom of this sooner rather than later (in order to avoid a complete loss of training (a partial reversal such as using diapers in daycare and on long car rides is probably fairly harmless at his age, as long as he is not upset by it)), though. Because of my own issues with loud noises I was not able to use any public restrooms until age five-and-a-half (almost six) in preschool and my mother only finished training me during that summer just before I started kindergarten. You probably don’t want to cut things that close when it comes to school enrollment in the next two to three years.