The riddle
How Do You Catch a Squirrel?
One word, a snack and a silly person
Squirrels eat nuts, and the word nut has a second life in everyday talk. Wiktionary gives the food sense as a hard-shelled seed or dry fruit. Its informal sense is slang for a crazy person, as in "driving like a nut". In the riddle it only means someone acting silly. A person who acts like a nut fits right in with a squirrel, and the question about catching it becomes a question about fitting in.
At home on a branch
Wikipedia counts about 285 living species in the squirrel family, Sciuridae, and notes that tree squirrels come down trunks headfirst by rotating their ankles 180 degrees, so the hind feet point backward and grip the bark. Each foot has sturdy claws for climbing. Anyone who climbs after one is entering its territory, which is why the riddle lets the person arrive among the branches instead of chasing from the ground.
Shade and tail
Squirrel comes from the Ancient Greek skiouros, a compound of words for shade and tail, a reference to the long, bushy tails many members of the family have. The same article lists seeds and nuts as the main foods, along with fruits, conifer cones, fungi and buds, so the answer matches the animal's menu as well as its habits.
Playing the nut
Put the question to the group and take a few guesses before giving the answer. Then invite everyone to show how a person would act like a nut: the best performance wins. The catch stays friendly, since nothing is ever caught or held, only imitated. Another animal riddle that turns on a single word: what do you call a fly with no wings.