In a collection of essays called “Virginibus Puerisque”, Robert Louis Stevenson writes the following passage:
http://www.online-literature.com/stevenson/virginibus-puerisque/10/
What is a “brown john”?
At a guess from context: a clay tankard such as would be used for beer. Never heard it called such though.
Huh. I’ve heard of an ale-jack, and since “Jack” is the pet form of “John”…?
A jack would seem to have originally been a leather tankard. Perhaps “john” was a hypercorrection? 
The OED also has an entry for a “Brown George”, which is defined as a large earthenware pitcher.
Possibly related to the word “Demijohn” (aka carboy Carboy - Wikipedia). Since a demijohn is huge, perhaps the character’s ability to drink is being described hyperbolically.
I guess a jug and that he has contempt for people with such, um, need a word, affectations?
Those are mostly showing up because of the maker’s name, John Dewar (hence a Dewar flask?) so I’m not sure whether this backs up the name.
Still, at least it didn’t return images of this kind of thing. 
I did an image search on “brown john” and maybe “whisky.” I believe those jugs are of the era relating to Stevenson’s time.
What I mean is that they show up because the description says, eg, “brown John Dewar jug”, rather than evidence of “brown john” being the generic name for them.
Swill means something like greedily imbibe. So this makes sense to me.
What’s the chick’s name?