If a Pokemon with Contrary gets a positive stat boost from a move that normally lowers them, does that mean if it used a stat-boosting move, its stats would be lowered?
Yes, it does reverse positive stat boosts. If a Contrary Shuckle uses Shell Smash, its Defence and Sp. Defence will be boosted and its Attack, Sp. Attack, and Speed will be lowered.
Source: This video at 6:10.