Green’s Dictionary of Slang

1. one who has round or bulging eyes.

1936		R.F. Adams Cowboy Lingo 179: We call to mind many seemingly mysterious monikers, yet each peculiarly fitted the individual [...] ‘Bug-Eye’, ‘Swivel-Eye’.
1955		‘Hal Ellson’ ‘I Didn’t See a Thing’ in Tell Them Nothing (1956) 129: Bug-Eye [...] He looks at me with them pop-eyes.

2. a round or bulging eye.

1978		J. Webb Fields of Fire (1980) 76: Snake shook his head in wonderment when he saw the bugeyes and the tight circle of mouth reappear.
1988		D. Woodrell Muscle for the Wing 131: L’il brother Francois concurred mutely, merely running his bug eyes to the nest and shaking his head.
2001		‘Randy Everhard’ Tattoo of a Naked Lady 8: I pulled out an old key I kept for just such an occasion. Dangling it before his bug eyes, I spieled how it was the key to her room at some motel outside of town.