Cut the tip off the caulk tube at a 45-degree angle with a utility knife. Slip the tube into the caulk gun, directing the tip through the front of the gun and positioning it so that the cut angle faces downward. Pull the trigger of the gun several times to move the pressure plate forward against the rear of the caulk tube, locking it into place.
Hold the caulk gun so that the tip presses against the seam between the joined boards in the foam box.
Squeeze the trigger of the caulk gun to start the caulk flow from the tip. Pull the tip of the tube along the seam, placing a bead along the seam to connect the two box walls. Release the trigger when you reach the end of the seam.
Moisten your finger slightly with water and then drag it along the caulk in the seam, pushing it into the joint and along the sides while smoothing the seam surface. Use the same process to caulk the remaining seams of the box.
Allow the caulk to dry overnight before working further on the foam boxes.