Green’s Dictionary of Slang

1. (Aus.) an eye infection caused by flies.

1895		Crowe Aus. Sl. Dict. 13: Bung, [...] swollen, as a bung eye.
1906		Sun. Times (Perth) 23 Dec. 4/7: A Bung Eye and the Curers Thereof. [...] A small specimen of insectivoria arenaceous, alias sand-fly, inserted his lance near my visual socket, injected a minute drop of poison therein and buzzed away [...] ‘You got a bung-eye. That’s wot you got’.
1957		J. Greenway ‘Australian Cattle Lingo’ in AS XXXIII:3 165: bung eye, n. An eye disease caused by bush flies, a particularly ubiquitous and noisome species, found only in Australia and India, that gather round the heads of men and animals. Stockmen and swagmen often wear cork-weighted strings dangling fom their hats to keep off these flies.

2. (Aus.) a black eye.

1983

J. Byrell (con. 1959) Up the Cross 45: Col also sported a flattened hooter as well as one bung eye and plenty of sticking plaster.