I was called upon by my daughter to remove a fairly large moth from her bedroom one evening, which I did, using the glass-and-card method - I did this primarily to avoid injuring the insect during capture and relocation - I'm not particularly squeamish about handling moths, but they will tend to damage themselves in trying to escape from a human hand.
As I turned to leave the room with the captive moth, I noticed two tiny, glittering brilliant pink jewels on its head - after a little bit of moving around and changing angles, it became apparent that these were cats-eye type reflections - from those compound eye elements pointing in my general direction, appearing when the light source was more or less behind my head.
I took a flash photo of the insect in the glass - here it is (and I promise this photo is completely undoctored):

So when insects get their photos taken with a flash, they suffer from red-eye, just like everyone else. Must be a bit of a nuisance at insect parties and social events.
What We're Seeing Here

At first glance, it may look like the phenomenon displayed in this photo isn't Red Eye at all, but just normal specular reflection from the surface of the eyes, if they were a sort of metallic red colour. They're not though - they're fairly dull, dark brown when viewed under ordinary lighting conditions.
What this actually is, is the reflection of light, back along its path of incidence. Moths have compound eyes - that is, they're made up of a mosaic of individual photosensitive units, each capped with its own individual lens. So each bright pink spot is actually composed of the reflections from a number of different eye-units - just the ones that happen to be pointing in this general direction on each compound eye.
Observing This Effect For Yourself
But don't take my word for it - you can see this effect for yourself in the comfort and safety of your own home.
Next time you see a moth climbing up the outside of a window at night, grab a portable light source and shine it on the insect from inside the window - if you place your head close to the light source, you should be able to see two tiny glittering spots of light reflecting back from its eyes.
