can you lose heat through your eyes?

The answer is yes, most definitely. I was a sharp shooter in the military and when we went out "night hunting", the enemy would camouflage themselves very well, thicker clothes, gloves and mud on exposed areas to keep their "heat signature" down. We had night vision goggles that we used to call "star light star bright" and the only thing the enemy could not cover up was their eyes. They would turn toward us and their eyes would look like headlights on a car. One round between those two headlights and we'd end that enemy's threat with extreme prejudice.

You can lose heat through any exposed surface.