n. — Food & Drink
a box formed of paper, polyethylene, and aluminum foil for holding liquid (juice, milk, wine), often with an attached straw.
Type: 2. Preservation — This term is a Canadian preservation. In noun compounds whose first element is an -ing form, the tendency can be seen to drop the -ing. This happened for, e.g. dumping truck > dump truck in the early years of the 20th century (Dollinger 2015b: 181-183). Some forms show regional variation, e.g. sailing boat in Britain, whereas sail boat is more common in North America (ibid.). Drinking box appears to be a Canadian preservation that used to be more common elsewhere, holding on to an older word-formation pattern with -ing (see Chart 1).
The term has much higher frequency in Canada than in the US, where drink box or some other terms, such as juice box, are more common (see Charts 1 and 2). This variable is one of the preservations in Canadian English that make some aspects of CanE more conservative than US varieties. See homo milk for another such preservation. In other aspects, CanE is more progressive than US varieties (see Tagliamonte & D'Arcy 2007: 70, Dollinger 2015a).
See also COD-2, s.v. "drinking box", which is marked "Cdn".
See also: fishboat
Images:

Chart 1: Internet Domain Search, 26 Nov. 2015 
Chart 2: Internet Domain Search, 26 Nov. 2015