Yours, Etc.: Origins and Uses of 8 Sign-Offs
c.1400, schurgyng, of uncertain origin. Perhaps connected to Danish skrugge "to stoop, crouch." Related: Shrugged; shrugging. To shrug (something) off "dismiss" is recorded from 1909.
a shoulder motion meant to express indifference, want of an answer, etc., 1590s, from shrug (v.).