You can wrap them with plastic, But I prefer the paper back insulation.
There is a slight chance that the plastic could hold moisture against the wood, resulting in damage. If you have trouble keeping the fiberglass insulaton attached or if you are just worried about it, You can stretch nylon string across it and fasten the string to screws on either side of the cabinets.
The foam is a bit like using plastic, if moisture ever gets between the wood and the foam, it may cause damage before it dries away.
Your cuts would have to be perfect to seal at the edges of the foam also.
Post a few pics when you finish. A lot of people would LOVE to see the finished product! This is a terrific use of wasted space that more people would be interested in if they could see how simple it is to do and how cool it looks and works!