Welcome! I'm betting your name is Jason.
I have that very quad. The reverse mechanism in the axle is actually from the boating world; it is the same system that switches an outboard prop to reverse. I don't think the problem stems from the reverse box, but rather from the CVT. Search for GY6 CVT on YouTube and you'll have hours of resources that will not only show you how it works, but also how to add cheap performance upgrades.
Which Province are you in?