An alternative etymology traces rush via Middle English rouschen (“to rush”) from Old English *rūscian (“to rush”) from Proto-Germanic *rūskōną (“to rush, storm, be fierce, be cruel”), a variant (with formative k) of Proto-Germanic *rūsōną (“to be cruel, storm, rush”) from Proto-Indo-European *(o)rewə- (“to drive, move, agitate”), making it akin to Old High German rosc, rosci (“quick”), Middle Low German rūschen (“to rush”), Middle High German rūschen, riuschen (“to rush”) (German rauschen (“to rush”)), North Frisian ruse (“to rush”), Middle Dutch ruuscen (“to make haste”), Middle Dutch rūsen (“to rush”) (Dutch ruisen (“to rush”)), Danish ruse (“to rush”), Swedish rusa (“to rush”). Compare Middle High German rūsch (“a charge, rush”). Influenced by Middle English russhen (“to force back”) from Anglo-Norman russher, russer from Old French ruser, rëuser.
Alternatively, according to the OED, perhaps an adaptation of Anglo-Norman russher, russer (“to force back, down, out of place, by violent impact", "to pull out or drag off violently or hastily”), from Old French re(h)usser, ruser (although the connection of the forms with single -s- and double -ss- is dubious; also adopted in English ruse; French ruser (“to retreat, drive back”)), from an assumed Vulgar Latin *refusō and Latin refundō (“I cause to flow back”), although connection to the same Germanic root is also possible. More at rouse.