In Walden did Thoreau use the term "animal products" in our modern sense?
I understand Thoreau's phrase to mean that these are animals that are products (of the environment), rather than products made from animals.
Rabbits and partridges are both edible, and it seems likely that Thoreau would have trapped or hunted them for food while living at Walden Pond (although possibly not, as he was a some-times vegetarian).
In another part of Walden, he mentions an unnamed visitor to his cabin who is a hunter, and mentions some of the things that could be hunted in the area:
[My visitor] would say, as he went by in the morning, “How thick the pigeons are! If working every day were not my trade, I could get all the meat I should want by hunting,—pigeons, woodchucks, rabbits, partridges,—by gosh! I could get all I should want for a week in one day.”
So rabbits and partridges were literally products (things produced by) the countryside that he (or others) made use of for sustenance.