Alt: squillgee, squillagee, squeegee
The origin of "squeegee" appears to be from "squeeze", although C.T.Onions, Oxford Dictionary of English Etymology, 1966, suggests that the C19th (second usage below as toggle) is perhaps a blend of "squelch" and "squeegee."
More often than not a mop for deck cleaning, etc. W. H. Smyth, The Sailor's Word-Book, 1867, gives both words:- "Squeegee: An effective swabbing implement, having a plate of gutta-percha fitted at the end of a broom handle" and "Squillgee or Squillagee: A small swab of untwisted yarn. Figuratively, a lazy mean fellow."
De Kerchove, International Maritime Dictionary, 2nd ed., 1948, gives "Squillagee: an instrument somewhat resembling a wooden hoe, with an edge of india rubber or thick leather used for scraping water from wet decks. Also called squilgee, squeegee" and "Squeegee: a strap with toggles in the end, used to confine a studding sail while being set."
Commodore Stephen Bleecker Luce, Text-Book of Seamanship, 1891, has a "squilgee strap" used for setting a stuns'l. At page 404:
The Topgallant Studding-sail. At sea, this sail is kept in the top, stowed up and down in the topmast rigging. To set it, order
Stand by to set the - topgallant stun'-sail!
Haul taut the topgallant lift.* One of the quarter-watch repairs to the topsail yard, where he converts the boom tricing-line into an "in-and-out jigger," and toggles the heel of the boom to a bull's-eye, which traverses on the jack-stay fitted for the purpose, or there may be a quarter-strap. (See Rigging Ship, p. 138.)
* It is observed that the support thus obtained is trifling. If, through neglect, the lift is not overhauled again after the studding-sail has been taken in, the yard itself will be endangered if the topgallant sail has to come in quickly.
[p405] The sail is cast loose in the top, having only a squilgee strap around it. Fig. 474. The halliards manned on deck, and the tack in the top, a hand by the sheet, and one also on the yard to assist to rig out the boom.