"As a jay," or often, "as a jaybird" is a idiomatic phrase used in English as an intensifier or superlative. It first appeared, as far as we know, in Middle English in the phrase "Jolly as a jay" indicating that someone was particularly cheerful.
Since that time, it has appeared in countless variations. Perhaps the most famous is the phrase "naked as a jaybird", which began as a variant on the British phrase "naked as a robin" (c. 1800s1). Jays were also common in similes at that time, as with "pert as a jay-pie" (1888) and "sweet as a jay-pie sang" (uncertain). The switch to naked jaybirds is usually credited simply as an American corruption of the preexisting phrase. There is another theory, however, that the phrase started out as "naked as a jailbird"2, which was later shortened to J-bird.
Other noted variants are "as crazy as a jay(bird)", "as smart as a jay(bird)", "as happy as a jay(bird)", "as free as a jay(bird)".... and so on, including "sober as a jaybird" (the top result on Urban Dictionary), "talky as a jaybird" (as quoth Truman Capote), and "queer as a jaybird" (John Henry O'Hara). Other potentially interesting mentions include the Ozark folk-song "Red as a Jaybird's Ass"3, and contrariwise, the American habit of referring to people as being as stylish or as natty as a jaybird -- as Americans associate jays with the striking American Blue Jay, rather than the less impressive Eurasian jay.
While this phrase appears in literally hundreds of forms, in multiple songs going back decades, and in books going back over a century, any given English speaker might be surprised and even confused to hear it. Specific forms are often localized to region, and it tends to have a folksy or old-fashioned feel. Use with caution, and be prepared to clarify.
1. A per the The Oxford Dictionary of English Proverbs. An exact date is not given, and the earliest record I have found is from A Collection of Poems by Stephen Fawcett, in 1872. It is perhaps worth noting that these phrases were quite popular, and included such entries as "as naked as a gorpim bird new hatched". There are no shortages of "naked as a..." phrases, from the quite sensible "naked as my nail" to the rather odd "naked as a needle". These were often used to throw shade, and one collection listed entries such as "as naked as a Strand Maypole", "as naked as Grantham Steeple", and "as naked as your Norfolk dumplings".
2. The theory here is that when people had their civilian clothes changed for prison outfits, they were forced to walk from the prison office to the showers naked before changing. As far as I can tell this origin story is completely made up.
3. Presumably related to the never-very-popular formulation "red as a jay-bird's ass in pokeberry season." The lyrics start "Her lips are red as a jay-bird's ass. Her eyes are cow-shit brown." It goes downhill from there. This was collected back in the 1940s, and lyrics and music can be found here.