Make a cube squish?

I’m trying to make a cube squish like a slime and then jump in BGE, all by pressing the spacebar.
The game logic is a simple thing, but everything I use to make the animation doesn’t work; Lattice deform is not supported, mesh deform works well only in the 3d prewiew and hooks don’t give the smooth effect I want.
I’ve also tried to apply the shape keys directly on the cube, but after that I can’t apply any modifiers on it.
Can someone please help me?

You’re wanting the middle section of the cube to expand outwards as the height of the cube reduces, yes? You can two this in a few ways.

I’d probably do this with blender render’s features, and then bake the transformation as shape keys. This is the same method I used to make waves in water. This method is hard on performance, but can look quite good.

The second way would be to hand-animate this with armatures and bone constraints. For example, having three armatures, one for the top, middle, and bottom set of 4 vertices. The middle armature’s scale is increased, and moves downwards, The top armature also moves downwards at the same rate.

I can give an example of the second one. The first one you can find a tutorial of on YouTube, then find a separate tutorial on baking shape keys.