How can I squash/deform a cube AND keep it positioned on the ground plane? My goal is to: 


Flatten out a cube 
Keep the cube's base on the same position of z-axis (i.e. flat on the ground plane)
Ideally, the volume of the cube should remain constant


Think of it like a block of clay slowly flowing out to the sides:  

(High quality 2d sketch:)



So here is my scene: I have a "cube" (black object) and a lattice (orange object) that encapsulates the cube:



Then I: 


Selected the lattice with B
CtrlP -> Lattice Deform
Added Shape Keys under menu Data: Object Data 


Then I tried to transform the cube with the lattice. Here is the result:




The way the cube is deformed is perfect, EXCEPT the base of the cube does NOT stay on the ground plane -- it gets pushed down below it.

So, my question is: how can I deform the cube like this, yet keep the base on the ground plane?
You don't need a lattice you can achieve this by only using Shape Keys:


Create two Shape Keys
Move the top face down
Scale the bottom outside edges outward