The OED s.v. dray offers “A squirrel’s nest” with the following in context citations:
1607 E. Topsell Hist. Fovre-footed Beastes 497 They..make their nestes, like the draies of squirrels.
1627 M. Drayton Quest of Cynthia in Battaile Agincourt 141 The nimble Squirrell..Her mossy Dray that makes.
The etymology of dray isn’t clear; it’s generally associated with the dray that means a sled or cart that lacks wheels, and is thus dragged. That dray derives from Old English dragan to draw; the OED suggests “compare Old English dræge drag-net, also Swedish drög sledge, dray, (Old Norse draga, plural drögur timber trailed along the ground)” (s.v.dray).
I suspect, though I can’t prove it, that dray for a squirrel’s nest also derives from OE dragan meaning to draw or drag, and refers to the way squirrels create the dray, by dragging leaves and brush into a nest in the fork of a tree. This is typically the way the North American Eastern Gray squirrel (Sciurus carolinensis) builds its nests.