I can see the issue you're describing. Still trying to track down why it isn't blinking instead.
Oh, wait, I think I know what's up here. Blink tagged layers show the layer tagged blink and hide all other members of their group (usually this hides the regular eye layer).In your case, the two blinking eyes are in the same group, so that means one of the eyes "wins" and the other gets hidden.
Small clarification. In Ch I see blink as both a group and a layer within that group. Because both the group and layer are named left/right blink, both are tagged. If you untag the two groups (leaving the layers tagged), the two blinks are each in their own groups, so they won't fight each other. For a more permanent fix, rename the groups so they don't have the word blink in them so they won't get automatically tagged in this way.