We just built/are building our coop, but it is in use now. We bought a standard cheapy screen door (the slamming kind I had when I was a kid, I love it). WE kept utility door closed til I could fix it, but I took 1/2 inch hardware cloth, used 1 x 2 or 1 x 3 slats my husband got from packing material at work (was in dumpster) and first stapled hardware cloth on to get it on there, then used screws every 12 inches to tack strips over the hardware cloth on to the wooden door to strengthen it. I put a latch just below the door handle that turns and "locks" as well as a slide bolt on bottom of door so nothing could "pull" the screen door open (it's not that flimsy, I am just being careful). Only thing is, I haven't used utility door since then, and found that I am going to have to notch the area where the utility door handle is as the screen door won't shut with the utility door shut (only will be a problem when winter comes, I will fix it before then). It worked great. Added lots of summer time ventilation.
Hope that helps, I hope I explained it well. Good luck. Also -- predators can come right through the screen door, also my bantums ate a screen that I had in the coop (had hardware cloth on other side so they didn't get out), but they lived and it was weeks ago.