Do all insects have exoskeletons

No, many invertebrates do not have exoskeletons. For instance, the octopus (a shell-less mollusc) is an invertebrate but has no exoskeleton - it uses it flexibility to move into deep crevaces looking for prey to eat. There are also earthworms (annelids or segmented worms) that are invertebrates but have no exoskeletons.

An exoskeleton is a rather specialized organ most commonly associated with insects.