Well he does have shifty eyes.

Can someone please explain to me what this term "swag" is supposed to mean? In Australia, a swag generally refers to your camping gear. I can't quite figure out what this new American slang term actually refers to.

No we didn't get it from your swagman stories, although the origin of the word is the same.
Swag refers to a thief's plunder or booty, though it has some other meanings today like free give-always. It is traceable back to a word imported into middle English from Scandinavia, swagga or sveggia which meant to sway, rock unsteadily or lurch. It evolved into that of hanging loosely or heavily, to sag. Its first use in English can be dated to 1520s. It was originally a bag of stolen clothes or the like and was English criminals slang for a quantity of stolen property in c.1839. It is not accidental in cartoons that a thief's swag is always shown in a sack. Thus and the Australian 'swag' derive from the idea of a bundle that sags or hangs down, often at the end of a stick. Swag also meant a bulgy bag in c. 1300 also of Scandinavian origin. It could also mean cheap and tawdry prizes at a fair.

It may have been invented by William Shakespeare because the word swagger, of which it is a shortened form, first appears in print in A Midsummer Night's Dream in 1590. Swagger means 1. A bold and arrogant strut 2. A prideful boasting or bragging. The rocking, swaying motion of the strutting links it to the Scandinavian origin.