First, I purposely didn't make this an issue in the Imagecache Javascript Crop (IJC for short) module because if it was a real issue, I would have found posts about it already, I'm sure.
I've installed/activated the IJC module and tried it "plain vanilla" - without Colorbox. It worked, but my client wants the visual improvements of Colorbox. Under "dependencies" on the Colorbox project page, it says:
Drupal 6 only: The Old 1.3.18 version of the Colorbox plugin and jQuery Update module version 6.x-2.0-alpha1 or later
I DLed the 1.3.18 version from the link provided and installed it in the appropriate folder.
I installed 6.x-2.0-alpha1 of jQuery Update.
From the IJC project page, I saw: "colorbox (make sure you included node/*/edit in the module settings)". The place on the Colorbox settings page where this applies actually says "Deactivate Colorbox on specific pages" (same as the wording in Thickbox, btw), which is a bit confusing in light of the "include node/*/edit" comment above. But, in any case, I've tried it both ways - with and without node/*/edit and node/add/*.
Nothing I've tried has caused IJC to use the Colorbox (or Thickbox) overlay instead of the default IJC popup. One anomaly I did see when I had Colorbox enabled was that, on the IJC default popup, when I moved the cropping frame to another part of the underlying image, the part of the image inside the frame didn't change, as it should. It just kept the same content as I moved it around in the larger original image. That problem didn't happen with Thickbox enabled. I suspect that there's a conflict of versions between jQuery Update, jQuery UI, Colorbox, and IJC, but I can't confirm that.
I've looked but haven't found any admin config setting(s) to tell IJC to use Colorbox/Thickbox, or to tell Colorbox/Thickbox to take the place of the IJC popup. Any help would be much appreciated.