I have always made the cleat from the same material the carcus is made from. This will keep the wood all of the same stability. In the case of veneered plywood. I have used both maple and oak since they are strong and the screw will have less tendancy to work its way into the cleat and becoming lose. I like to make the cleats 1" or 3/4 x 1 anyway.

I have not tried it yet, but I hear some people like the Gorilla Glue. suposed to be good for end grains in corner joints and the like. I think a real problem is you need about 5 years of expanding and contracting with the season changes to be able to see how a joint is going to hold up.
To offer a suggestion of yet even more work! put two coats of finish on the inside of the box too.

Bill