Rodents have a muscle connected to the jaw that runs through the eye sockets, so all I can think is that action of chewing engages that muscle and causes his eyes to become more prominent, similar to how rats' eyes boggle when they grind their teeth.
This article talks a bit about the facial muscles:
Mouse-Like Rodents: Myomorpha
That doesn't explain why it's only recently been happening though and I'm not sure why that would be.