Can Final Cut Pro do White Board Animation?

You can't really do that in FCP. You should use a motion graphics application like Motion or After Effects. The white board designs have to be created in graphics applications like Photoshop or Illustrator.

It depends on what you have in mind, and what exactly what you mean by whiteboard animation. If are filming someone drawing on the whiteboard and want to edit this, fine - maybe do a timelapse, apply visual effects, and so on.

On the other hand if what you have in mind is creating a computer generated animation then that is not what FCP X is targeted at. You may br able to do it, but you’d better use an animation program. You might still want to use FCP X to edit the final video.

Perhaps if you can elaborate a bit more someone here can offer a more tailored advice.

If the animation is not overly complex, you may succeed in doing it with FCP X.

You may for example prepare one or more layered Photoshop files and animate the layers in FCP X.

Can you point us to an example video that is similar to what you want to achieve?