Can whatever it is get into your shed easily, ie without having to gnaw its way in?
If so, then it could be either.
The droppings look a bit more like squirrel than rat but if there is no easy access to the shed I'd be surprised if it is squirrel and would lean towards rat. We have squirrels in the garden everyday and also have sheds but none have been gnawed into.
However, both are easily capable of doing that sort of damage to plastic storage boxes.
Dave