Either way you say it, American with w or British with y, the names “Crawfish” and “Crayfish” are derived from Old French crevice or crevisse, meaning “crab,” a term that then covered almost any of the larger crustaceans.
And because of the propensity of the critter to navigate rearward, we have made a verb of the term, using it figuratively to indicate an attempt, usually conspicuous, to withdraw oneself from a stand or a commitment that has become undesirable.