The first thing you should do is find an expression of the volume of the box in terms of the constants a, b, and c.
One suggestion I have is to use the volume integration method to get the "slice" of the volume that is not inside the region of the box and relate that to the constants a, b, and c.
To make it a little clear picture a two-dimensional analog where we inscribe a box in a circle.
Now consider that we are finding the areas to the left and the right of the box in a 2D sense.
The area of the box + area outside box = area of ellipsoid.
If we choose a width say t units and integrate from (-a,-a+t) for f(x) = 2*SQRT([1 - (x/a)^2]*b^2) [We use the 2 to get the area bounded by the lower curve] that gives the area to the right of the box.
By changing limits to (a-t,a) we get the right volume.
You can then obtain the area above and below the box in a similar fashion and relate the volume to this magical t value.
Once this is done you are maximizing area against this t value and this is just a normal optimization.
Now extend this idea to three dimensions by changing areas to volumes and you should have something to go off.