The headset cups will likely be different on the modern kids' bikes, but the one piece crank stuff will drop right in...the catch is, you'll won't be able to rn the Chicago Schwinn cranks (if you still have'm) as those are threaded differently (28tpi for Chicago Schwinn vs 24tpi for almost everything else...) Even the cups are different, b/c they're ground for a different bearing set-up, or so I've been told. I never tried to mix'n'match the old Schwinn cranks/hardware/bearings with the standard cups, but Schwinn OPC BB's do take a different retainer than the standard cranks do....
As for the headset, it's tough to beat a real old-school Schwinn one, but there are plenty of "BMX"-spec headsets that will fit the 32.5mm headtubes of a Chicago Schwinn. Tioga Beartrap II is a good one, but I like the generic VP units that you can get from porkchopbmx.com....
You'll want a 13/16" seatpost; get one with 7/8" tops so you can run standard/normal guts....
Basically, the rest of the bike will take standard parts, although not every 1" seatclamp will tighten down sufficiently on a ChiTown Schwinn.