I think you need to add a second face behavior. In the puppet panel, select what you want to be your "second face," and in the behaviors on the right, click the + and add Face. Now when that second Face behavior is armed to record (red dot is on), it should also start picking up your eye data. It's kinda like how you have to add a second dragger behavior currently if you want to control multiple draggable points independently (see Sticks & Dragging (Adobe Character Animator Tutorial) - YouTube).
So when I add a second face behaviour the eyes still don't move. I tried to make a complete alernative head and added the face behaviour on that one. Now the alt. head itself move and so do the eyebrows, however they eyes still aren't moving. Also the original head isnt moving anymore, except the eyes.
I feel like there should be a replacement behaviour. When activated it would replace the desired part with another texture. Like in my case the pupils. Probably would be helpful for other projects as well.
Hmm. If you want one of us to take a closer look, feel free to share the file and we can try to troubleshoot it (and/or post a screenshot of your layer setup).
Currently things like "Face" and "Lip Sync" are looking for the first instances, from the top down through the layer list, of applicable names. So like in the Cyberman example from the last live show (https://youtu.be/H4J9tKuibVk?t=19m16s), at first it was only recognizing the top face / lip sync, but after applying more behaviors to the lower ones it started to work. Maybe it's an ordering issue?
One other possibility - for eyes to move, they need an eyeball, which sets the bounds for where the pupil can move in. I've done a workaround with this sometimes for some of my characters who don't have big bulgy eyeballs by making an invisible eyeball layer, either one with 1% opacity or copying an eyeball shape from the face background texture/color.