In this case we always use "折れる".
折れる is used when something long (like a stick, bone, etc.) gets broken into shorter pieces.
壊れる is used in a broader sense to describe "to get broken, destroyed, wrecked"
BTW, another useful verb is 割れる, which means something round or flat (like a dish, egg, etc)
gets broken or something long gets chopped (split) in two pieces (cf. chopsticks, 割りばし)