In addition to using something waterproof-durable-flexible as flashing over the hinge, you could also use that (or wood I suppose) to make long flaps or sides going down from the 3 free edges of the nestbox lid. I think you'd want them going as far down the sides of the nestboxes as is structurally-stable, stopping just short of either a) wibbliness or b) their getting in your way when you open the lid and want to reach in for eggs. That should defeat an awful lot of your sideways-blown rain.
You are likely to get SOME leakage into exterior nestboxes no matter what, though, unless they are in a very protected location. That is just a basic downside of the arrangement. If your coop is high enough, you could consider rebuilding the nestboxes on the interior of the coop (high enough that chickens can still walk under them so you are not losing any useful floorspace, but still lower than your roosts) with just an access hatch flat against the coop wall. You will still have to worry about flashing/gasketing/etc but at least whatever bit of weather blows in will not soak the nestbox nearly so badly.
Good luck, have fun,
Pat