Thanks for the warning. Is there some sort of material that can be sandwiched between MDF and solid wood to mitigate this issue?
What WGH is saying is true, but there may be a way to use the panels you ordered.
If the panels are 26x14, (a photo would help) I'm assuming the grain of the wood is the 26 inch dimension. There will be very little humidity caused expansion in the grain length...so little that you can pretty much ignore it. The grain width, 14 inches, is where some expansion will occur. In 14 inches there will be very little expansion. As an example, I built my kitchen table. It's 42 inches wide and 1.125 inch thick solid (curly) cherry. I built breadboard ends on the table to accommodate the top width expansion. Throughout the seasons(winter, forced air heating, very dry) the top expands and contracts about 3/16ths inch on each side. So that's 3/8ths total expansion over 42 inches.
Now, if that 42 inch wide table top was glued cross-grained to the breadboard end on one side, and thereby inhibiting or stopping expansion on that side, but the other side of the top was allowed to move with the seasons, then that one side would have about 3/8ths movement over the seasons. The top will expand and contract like normal, but only in one direction.
Ok..now for your 14 inch wide panel..we're looking at
about 1/16th inch total potential expansion, or about 1/2 of an inch on each side. If it were me, I'd apply a couple coats of finish to all six sides of the panel. This keeps moisture entering and leaving all sides at about an equal rate. Then I'd mount the panels on the mdf sub-box sides with a half dozen or so dime-sized dabs on RTV silicone(or pure silicone caulking..basically the same thing). Apply the dabs to the wood panel (a couple inches from the edges so it doesn't squeeze out past the edges) and lay it RTV side up on the floor. Tilt the sub onto the panel and line it up the way you want it. Let it dry for a day. Then do the same on the opposite side.
The RTV-silicone will hold the panel to the mdf box and, with RTV remaining flexible, will allow the wood panel to expand as needed. The wood will expand/contract, you won't stop it, but it won't crack. I'd expect, if the panle was made correctly, you won't have any bowing issues..post a photo of the ends and we can take a look.