According to Keeping Insects, a stick insect eats leaves. Every species of stick insect does not eat the same kind of leaves. If they are fed the wrong kind of leaf, they will starve.
Some of the types of leaves that stick insects eat are bramble, oak, hazel, privet, rose and ivy. The leaves must be fresh. Take cuttings from the plants instead of tearing the leaves off. Place the cuttings in a cup of water to keep them fresh longer. Place sand or stones in the bottom of the cup to keep the cup from tipping over. When the leaves start to dry out, replace the leaves with fresh ones.