The answer to this question has to be fairly vague. Paulsen published the original Hatchet book in 1986 and set the book in modern times. Readers are not given a specific date regarding when the book supposedly takes place, but we know that the story is meant to take place in "present day," as Brian is flown to his destination in a modern Cessna aircraft rather than some DC-3 from World War II. The River was published in 1991, but the events of the story take place two years after the events of Hatchet. Brian survived for fifty-four days in the Canadian wilderness in the first book. The government wants to know how he managed such a feat, so they ask him to lead a trip back into the wilderness.
Other than taking place two years after the original story and being somewhere in the Canadian wilderness, Paulsen does not give readers much more setting information. We are told that Brian and his companion are by a lake with a river flowing out of it and that 100 miles to the south is a trading post. That gives readers something to visualize, but it is ultimately useless information, as that could be dozens of places in the Canadian wilderness.
This is a good thing for the story. We want to feel just as lost as Brian. His lack of tools is also a good thing for the story. First, it shows us that Brian is truly dependent on rudimentary tools and what he finds in nature. Second, those rudimentary tools give the story a timeless feel to it. Things like knives and hatchets have existed for hundreds of years. Paulsen doesn't give Brian any piece of technology that would allow a reader to say that the story is taking place in one time rather than another.