I think that would mean only a dyson hemisphere, since Earth is illumnated by the entire hemisphere of the sun that faces us, due to the sun's sheer size.
To see the kinds of effects this would have on the gas giants would be fascinating. It would probably drastically alter the coriolis banding and spots on Jupiter and Saturn.
Venus would probably be the only other planet affected, and perhaps comet behavior might change. In fact, I wonder if planets would become so cold, that they might produce visible comas as they warm up, when they enter the sun's (now hemispherical) light cone. That's probably the most prominent effect I can see happening.
But does it have to totally block it out? What if we left space in between units so that it just dimmed the sun behind it. Hell, maybe it will help reduce global warming so we can just keep on keepin' on. No need to reduce output levels of anything other than the amount of sun we receive.
Plus, it would be fun to screw with societies in other parts of the galaxy that are using their Kepler satellite equivalents. Every time the Earth transits, the light would dip in a way that would lead the observers to think Earth was much larger, thereby much less massive (for the math to work). Great defensive strategy as they would not think our planet was worth pursuing.