I can assure you that there is very little design commonality between JWST and surveillance satellites. JWST is a very unique mission that is completely different from that of spysats.
This is probably unlikely as JWST is designed to fly in L2 and not in a geosynchronous orbit. Additionally, at the wavelengths it is operating at (mid-infrared), if it pointed towards Earth, it'd just see the heat of the Earth's atmosphere. Probably not very great for spying.