[psa]

I've used self-leveling watering trays in the past with an inverted watering container. Water coolers and automatic pet water dishes work on this principle. When the water container is turned upside down with the opening submerged, no water comes out because of the vacuum in the container. If the level of the water falls below the opening in the container, air is let in and the water is replenished until the level is above the opening again.

I've also used 55G plastic water drums with an opening plumbed at the top and another at the bottom, each with a valve. Open the top and close the bottom to fill. Close the top and open the bottom, and it can feed the water tray.

Back to the topic at hand-
If you set the water container/drum at a higher level than the top of the earthbox, you can run a hose from the bottom valve of the container down the watering pipe of the plant box and it should be able to maintain the level of the water at the same level as the bottom of the hose, with the following considerations:
--the hose used must be wide enough to allow bubbles to pass upward at the same time that water is moving downward (this is a surface tension consideration that will also depend on the material of the tube)--no 1/8" tubes here
--there must be no point between the bottom of the hose and the entrance into the main chamber of the water container where it is not on at least a slight incline, or bubbles may become trapped and shut off the flow of water. In practice, I've made this kind of thing work with a horizontal bulkhead/valve on the bottom of the tank, but YMMV, and a wider outlet there will have less trouble.
--air must be allowed to enter the water chamber in the plant box, either around the tube in the watering pipe or through the overflow hole (I don't trust small overflow holes because they have a tendency to get plugged on me just when I need them most).
--cutting the bottom of the tube at an angle may help start the flow of air up the short side of the tube and water down the other side, at the expensive of allowing the level of the water to vary a bit more.

Finally, I haven't actually done this with an earthbox-like container. I have used it on wicking trays, humidity trays, and small ponds with good success. I will probably give it a go if I ever spring for an earthbox or try making one.