That'd be awesome if you could prove it to me mathematically so I could be sure. Everything they've always taught me in classes seems to say you're right, but there's something about it that's nagging me.
It seems like if you had a mass on the end of the spring and let it drop... it'd occilate for a while, but eventually stop due to the energy lost each time. But if you continued to move your hand up and down at the resonant frequency it would move forever. With the little "machine" I designed you keep getting a "boost" due to the casimir force pushing at the "resonant frequency" of the machine if it was designed that way. Also, after reading up on it some more it seems that at seperation widths greater than 1um you get a bigger "push" together due to thermal fluctuations. furthermore, there are two forces that depend on the seperation being at "cavity-resonance". At a cavity-resonance frequency the radiation pressure inside the cavity is stronger than outside and the mirrors are therefore pushed apart. Out of resonance, in contrast, the radiation pressure inside the cavity is smaller than outside and the mirrors are drawn towards each other. Since, in reality, the attractive force beats the repulsive force we could have two mirrors at the cavity-resonance seperation which would cause them to move in and generate your current. Due to the repulsive force that gets bigger when you're out of "cavity-resonance" you wouldn't have to generate the same amount of energy to push them apart. You'd get some extra help from the casmir repulsive force and from the fact that on charged plates the force generated is WAY stronger than the casimir force.
I can see you starting out with a device manufactured to start out with a speration greater than 1um and at "cavity-resonance" so you get a big push together and generate some current. Then, once they move in together, your plates get charged and the respulsive force from the charged plates at such a close distance is enough to beat out the casimir force. Especially since now, at the closer distance, the force pushing them together from thermal fluctuations disappears at distances less than 1um and and now that we're out of cavity resonance we have less casimir force pushing inward. Also, we have momentum to help carry the plates back over the 1um seperation mark.