1DX Memorize Focus Points

I know that 2 can be memorized ie, horizontal and vertical but can a third be added in the horizontal or vertical mode by assigning a custom button?

Yes. It is called a registered focus point. I register a focus point and use the * button to activate it sometimes.

To register the AF point; press the AF Point Selection button and ISO button simulatenously.

To un-register the AF point, press the AF point Selection Button and the +/- exposure compensation button (which is next to the ISO button) simultaneously.

Somewhat related, but with the 5D3...

It drives me nuts how it "remembers" the AF point between varying AF modes (spot, expansion, etc...). As I sort through the types of AF, it will move my AF point back and forth. Not sure how or why it does this.

Anybody care to share? :P

Different things are mentioned here.
First, there's the orientation linked setting. When enabled, the camera will remember the AF point last chosen in three different orientations: Horizontal, vertical with grip up and vertical with grip down.
It will not only remember where the point is, but also the AF point selection mode. Thus you can go with single point with expansion in horizontal but have no expansion in vertical, for example.
When this function is enabled, you simply set the camera up by holding it in a certain orientation and do your settings. They are stored automatically.

Then you can store a registered AF point. That's something else, as it becomes a home point (HP) you can return to, regardless of how you've moved around the selected point. You can store an HP high up in the image, and then program the camera to use that by pressing a button, regardless of where the selected point is. This is the point that's stored by pressing the AF point selection button and the ISO button simultaneously.

Now if you combine these two, you can store your home points at different locations in different orientations. Store the HP when holding the camera in a certain orientation.

Don't know about the 5D Mark III, but the 1DX remembers the same location of a single point across all four single point selection modes. Then a zone can be at a completely different location.