Much the same you would care for them as an adult. Nymph stick insects are just like smaller versions of the adults; they eat the same leaves (although for very tiny hatching's it is advisable to trim the edge off the leaves so it is easier for the stick insects to eat), and live in the same environment as the adults. Be sure to spray the enclosure the nymphs are in extra lightly as very small nymphs and hatching's can easily drown in water droplets.
Mastervenusflytrap: I recommend putting them in a plastic container with holes on the top, also give them the new eucalyptus shoots only they wont eat the big leaves they are to hard to eat.
Also if you put the nymph's in with the adults it is a bad idea because the adults will crush them if the nymph's climb over the adults.
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You need a suitable container, one that is tall more than it is wide. Stick insects need height more than length, due to stick insects moulting vertically. Size of the vivarium or tank is very important. It needs to be three times as high as your insect.
Some species like arid environments, while others like very humid conditions. Find out what your species likes. You can get pre-built vivariums that have mesh on some of the container sides. Most are completely made out of glass.
You will need a source of food (also how you are going to keep the leaves fresh - such as putting sprigs in a narrow-necked pot) for your stick insects, and a water sprayer. Stick insects like to drink, so spraying the leaves every evening allows them to drink and help keep humidity levels up. Some species might require extra heat, so a heat pad may be needed.
Lastly, some stick insects need soil on the bottom of the tank, as the females prefer to bury her eggs in the soil. However, most will just drop the eggs and need no further help.