The expression is most used to describe being naked in a public setting and without embarrassment. So during the 1973-74 fad of streaking, my college roommate sophomore year at Miami, had the courage to streak at dusk one fall day in South Quad. He was naked as a jaybird. Ray Steven's in his song "The Streak" warns Ethel not to look at that feller - nekked as a jaybird - nothing on but a smile.
How a jaybird is naked - I have no clue. An internet search provide no definitive answer - theories about the word "jay" as slang for hick seemed the most plausible to me. After all who else says nekk-ed.