A chinchilla has a natural defence against predators ... it can allow it's fur to "slip" ... leaving the prey with only a mouth full of fur and no dinner.
Chinchillas, as a rule, are not 'hold me' animals ... they love to run and play. They can, with their body temperature + your body temperature, easily become over heated and get very uncomfortable.
Another reason they may begin to squirm, is that they may need to 'go tinkle' and they do not want to urinate on you, so they are trying to ask you to let them down.
Being held improperly is another reason they may try to get down/loose ... if their legs/feet are not supported
at all times, they feel like they are falling and are trying to save themselves.
If you were stuck in your room 23 out of 24 hours every day ... would you want to set quietly when you were allowed to come out of your room?
Jo Ann
P.S. If one of my little ones were to sit still when let out for play time ... I would think he/she was sick.