1/4" glass is plenty thick. 14x14x14 cube needs 6.3mm glass on the bottom and 3.8mm glass on the sides. (1/4 inch = 6.3mm) according to my calculator. I assume the bottom is assuming that it's going to be within a frame, and not simple just glass sitting on a flat surface, but I'm not sure.
The corners obviously aren't going to have a glass on glass joint. I've seen someone do that with a 6x6x6 tank. Someone (I believe it was hoppy) mentioned that the majority of the strength of the silicone joint is on the glass on glass contact, and the shear strength is very low. I would be hesitant on building a tank and relying on just a corner fillet only. You might be able to cutoff 1/2" from the sided pieces, so you can assemble it like a traditional aquarium, with the glass on glass contact at the corners.
Use aquarium silicone. I don't think I've seen any other adhesive recommended for this job.