Good question... The best way to think about the two sentences is in terms of when the action happened or should have happened.
1) She was supposed to move boxes.
"was supposed" is the simple past tense, so the action happened in the past.
"to move" is the infinitive, but it too is a single action.
Here are a couple of similar examples:
She was scheduled to do something.
She was told to do something/ She was told to move the boxes.
So, "was supposed to move boxes" means that in the past she was supposed to do something. The time of the action has passed and is over/finished.
You were supposed to cook dinner for your friends (but the time has passed).
2) She was supposed to be moving boxes.
"to be moving" tells us the activity or action was supposed to be occurring/happening at a specific time, and over a period of time (continuing action).
She was supposed to be moving boxes, but she was talking to her friends at the time she was supposed to be (scheduled to be) moving boxes. So, "supposed to be doing something" is referring to the period of time when she was expected to be moving boxes, and the first sentence is referring to the fact that the action (moving boxes) did not occur in the past.
OMG... I wonder if this explanation is clear and helpful... I hope it helps...