Short term solution - Cut off the caulking, remove the nest box assembly, remove all the screws/nails from the siding at the top of the opening so you can slide flashing under it, form flashing to go under the siding about 3" then over the horizontal section and drain over the flexible hinge.
Better long-term solution - rebuild the nest box with a sloped roof that starts at the coop wall and has a 2 in overhang over the front wall. Then hinge the front wall so it drops down for egg collection. Make sure that you have at least a 3" lip at the bottom to prevent nesting material from falling out every time you open it up to collect eggs. When fabricating the box make sure that the front opening will clear the box roof overhang before you install it. You would need to install the flashing as described above to prevent the water from getting into the box. I would also put a good quality, double coat of exterior grade paint on the sides, front and bottom of the box and roofing material on the roof. Roofing would go on first with the flashing extending about 3 in over the top of it. OSB was not meant to be left exposed to the elements.
