I went ahead and tried some new approaches to solve the glitching and seam, based on recommendations by @mspeer.
Here is what I did:
- Mesh Resolution - Experimented with changing mesh resolution to see if it would give smoother results.
- ShrinkWrap deformer - To compare results of a deformer to Boolean alone.
- Textures - Texture blending at the seam with the Boolean operation.
Mesh Resolution
This was my mesh resolution in my previous attempt.

I wanted to see if lower resolution would give smoother results, so I tried this.

Here are the resulting topologies after the Boolean and smoothing.
High

Low

And here is the resulting animation:
High
https://streamable.com/lxh78
Low
https://streamable.com/lby54
So which is better? There are two problems I'm trying to solve: 1) smooth movement and 2) beveled seam (surface tension effect).
1) Smooth Movement
High res approach looks better. The movement of the low res option is less desirable. It's a lot more jumpy and glitchy. I want it to move as smoothly as possible, like a fluid liquid surface.
2) Beveled Seam
Neither of these approaches seem to give me any control over the beveled seam. The Low Res solution does look a bit more soft, but when it moves, the triangulation becomes very obvious and doesn't work for me.
ShrinkWrap Deformer
I wanted to see if the issues could be solved instead using a deformer. I went through the deformer list and the only one that I found might work for me was the ShrinkWrap deformer.
So I made a second cube and wrapped it to the boolean.
Before wrapping

After wrapping

Here is the resulting animation:
https://streamable.com/xbabb
1) Smooth Movement
The results are pretty good. I can see a certain "voxel" effect as the geometry moves across the surface. Perhaps it would be smoother with more subdivisions.
2) Beveled Seam
I noticed that the ShrinkWrap effect has a falloff and inflation setting. I did not play with these very much, but perhaps they could give me some control over the beveling.
Here is the topology and inputs.

Clearly, I now have a new problem with the ShrinkWrap that I did not have before. The ShrinkWrap deformer does not allow for "undercuts", which I would need in this case, in order to get both the front side and back side of the arm that comes out of the box.
Textures
I did a quick experiment with textures to see how the two textures would blend after the Boolean operation.
Here are the objects with textures applied.

And here is the resulting animation.
https://streamable.com/zdeck
Ideally, I would like to find a way to control the transition from one texture to the other, using a smooth ramp or something to that effect.
In summary, here are my questions:
- Smooth Movement - How to create smoother movement when animating a Boolean union?
- Beveled Seam - How to control the softness of the beveling at the seam on the Boolean union?
- Deformers - What deformers should I consider looking at to solve the problem? (must allow for undercuts)
- Textures - When using a Boolean, is there any way to do a smooth blend of the textures two objects (other than in compositing)?
- Voxels/Particles - I would like to consider a voxel/particle approach. What tools in Maya should I look at?