This is impossible to „diagnose“ via the Internet.
The list of what may be wrong is nearly endless, but some of the more frequent issues are, in no particular order:
- Internal / organ damage.
The pups are extremely fragile and both it falling from the nest and
your dog carrying it around may have caused harm.
- Thermoregulation problem.
This pup is probably less than a week old and at that age they require external support to keep their body temperature in the appropriate range. Deviation can cause all kinds of issues, I am not that familiar with palm squirrels, but bat pups (even smaller mammals) won’t eat if they are too cold.
Like many small pups, squirrels need gentle(!) massaging of the belly to pass stool.
- Aspiration pneumonia.
Aspiration of milk can happen quickly as the amount of milk the pup can swallow per gulp is so minute. If the milk gets in the lungs, it can result in pneumonia.
- Random infection.
As the immune system of the pup is still weak, the baby can catch any random virus/fungus/bacteria, making treatment necessary.
Please contact a vet in your area to have the pup examined properly.
A slight constipation or low body temperature is something you can handle at home (and I recommend you do immediately), but everything else needs proper treatment.