Like you, I use a big box and a run. I have it indoors (the garage) until they go off the lamp at about 3-4 weeks for ease of electricity supply. I use a red bulb as it's kinder on the chicks - it's easier to sleep under it than under a white bulb. The light's on a chain and I raise it up and down depending on the chicks - if they are huddled together they are too cold and I lower it, if they are all spread out, I raise it. I keep them in the box - it's about a metre square for a week or so before letting them into the run and I shut them in under the light at night, with food and water. Don't underestimate how small a mesh they can get through or how easily "lost" they get when they are tiny. If they wander too far from the light, they can't find their way back and die.
Good luck with your hatching - it's great fun!