As Daniel noted, the fret board is usually flush with the box top or raised somewhat above it. If I see your photos correctly, you have your neck entering the box below the box top. If your neck is removable, you can put a step in the neck that is the same thickness as the box top thickness. That will make the top of the neck flush with the top of the box. If you add a separate fret board, it will be higher than the top of the box (that's a good thing).
There are a number of books on building box guitars. Just search amazon books. Here is one good book
Good luck with your builds..
Tim, brose the "photos" on the home page, see some examples. we need more photos, maybe one of the top. it looks like the top is a solid piece of wood. if it is, and it was me, i'd cut the top off, flush with the fretboard. crazy, i know. if that's an option, i can give you a few ideas.
Here’s some simple videos about notching the neck to fit the box
From my limited experience, I tend to like having the fretboard higher than the body. It gives you a bit more choice when it comes to bridge height so that you can fine tune the string action.
And if you plan on using pick-up, you definitely need that space.
Looking at your pics, you'll need a pretty thick nut and you'll be limited to playing slide as you won't be able to push down the strings so that they make contact with the frets without hitting the body. Also, when the string action is too high, the notes will pitch up more than you expect when you press down on the string.
Hi Tim P, if the photos you show are of the guitar you are building, then you may find that just adding a fretboard on a neck that low into the box, will still leave you with a very low bridge. As Tom T states the bottom of the fretboard is better over the top of the top of the box lid. A good bridge height is 1/2" to 5/8" including the saddle.
Looking at the pictures, I believe you reclaimed a guitar neck. I can't tell if the neck extends all the way through the box.
But either way, if you want to keep the frets, you need to shim the neck from the bottom of the box and maybe cut the box a bit so that the fingerboard sticks out on top.
I know it's a bit more work but it sure beats re-fretting an entire neck!