Can the spirit companion move anytime the shaman takes a move action, or does it need to be a move action where the shaman moves?
As per the Q&A section in: linky
It moves up to your move speed when you take a move action
Thus, as it explicitly notes a "move action" it only moves when you take a move action. However, it can be any move action. Take a look at the Dragon Article they reference for more details. Unfortunately, the wording here is pretty unambiguous.