How to add "bend deformer" inside After Effects?

You can try the bezier warp tool.   You'd have to have the square in it's own composition that matched the size of the shape.  back in the main comp move the precomp's anchor point to the correct spot and apply the bezier warp.  key frame the verts and tangents of the warp tool along with the scale of the precomp.  Precomp's x & y scale should be unlinked so you don't have uniform scaling.  You have to really pay attention to using hold keyframes too because you'll be making lots of adjustments and if you key something that doesn;t have a hold frame applied it will mess up the tranformations. I've never tried this before but there's a plug-in called Squash and Stretch by Mamoworld and I think that's how it pretty much works aside from the interface.  Once you get everything keyframed you can save it as an animation preset.  I'm messing around with the bezier warp as I type this and it's pretty cool, actually.   Once you have the effect applied just scrub the values in the effect controls panel to see what they do.  You'll get the hang of it no time.  

There are several approaches.

If the box is a shape layer, you can animate each path anchor.

You can apply Puppet Tool, which will deform the box according to the pins motion.

And there are actually bend effects in the effects panel, with a similar outcome to the gif. 

You might need to combine several effects to get the same result. For example, when the box touches the ground again, it's rather a simple scale animation then a bending. The bounce on the other hand can be done with expressions, instead of keyframe animation.

Best is if you start with puppet tool. This gives you the most freedom to shape your animation.

*Martin