In the Jeju dialect, 독새기 means "egg" (계란). I have heard that the word comes from 닭새끼, which means "baby chicken."
I am not really into dialects, but if I lived in Jeju-do, I might spend some time studying their dialect. For those interested in the Jeju-do dialect, here is a page that lists some terms and expressions. And for those interesting in reading some old Jeju-do stories, here is a site that gives you the story in the Jeju dialect, along with a translation in standard Korean. The stories are short, so for fun, those of you who live in Jeju-do might want to memorize one in the Jeju dialect and tell it to one of your Korean friends there.