Are all animals with exoskeletons vertebrates or invertebrates?

An exoskeleton is not an animal, but a part of an animal. A skeleton is simply something in an animal that provides as a structural support and (except for sponges) something to anchor the muscles to. There are two sorts of skeleton:

• Endoskeletons, as the name indicate, are part of the animal's interior. This includes a human's bones, which are inside the body. They allow for much more flexibility and do not need to moult their entire skeleton at once.
• Exoskeletons, on the other hand, are on the animal's exterior, and serve as a sort of armour. The arthropods, which include the insects, crustaceans, millipedes and centipedes, arachnids, trilobites, sea scorpions, and many others, all bear chitinous exoskeletons.

The two types of skeleton are not mutually exclusive; an organism can easily have both. The first example that pops into my mind are the ostracoderms, which lived from the Lower Ordovician to the Upper Devonian. These were jawless fishes that had armour on their outside for defence, but also had vertebrae and an internal skeleton.

So, long story short, an exoskeleton is not an invertebrate, nor is it a vertebrate. It does not meet the criteria by which this is determined.