I couldn’t find “jay-walking” in this 1909 dictionary, but I did find “jay,” and it doesn’t necessarily refer to a blue jay:
Jay, jā, n. a bird of the crow family with gay plumage: a wanton woman: an indifferent actor, a stupid chattering fellow. [O. Fr. jay (mod. Fr. geai); from root of gay.]
From Chambers’s Twentieth Century Dictionary (part 2 of 4:E-M), 1909. Ed., Thomas Davidson.