You can't do this in AE using the standard set of distortion effects.
Cinema 4D Lite only includes a few basic deformers. Best no budget option: http://blender.org. A very powerful but a little clunky open-source 3D app that will do the trick.
The first and probably simplest to use third-party plugin I can think of that would kind of do the same thing is FreeForm Pro | Mettle. It's a couple hundred dollars but it is pretty easy to use if you need deformation effects and their support is great. There are also other solutions but for deforming a grid so that it looks like a cube is pushing on it but I think the FreeForm Pro plug-in might require a little less training and fiddling than other options.
Another option may be Element 3D V2.2 from Video Copilot. I know there are deform tools in Element but I'm not sure how or if you can make one object deform another. It might be worth exploring if you have Element or intend to do a lot of 3D work in After Effects.