This issue has been resolved. Long story short, I uninstalled Adobe Air from Applications/Utilities/Adobe Air UnInstaller. I then proceeded with #2 above. I then went to the Adobe Air forums and downloaded a much older version of Adobe Air, Adobe air 4.
I then reinstalled JuiceBox Pro version 1.4.3 and all is well. There was a corruption in Adobe Air installs at some recent point that was causing this problem. I believe it is now safe to upgrade to Adobe Air 16 as it is working on my other 2 computers.

Steven,
Thank you for your response. I have already tried this several times and have yet again after reading your response. This is just not working. I even am using the download file from my computer that is working and installing 1.4.2 as it was working fine previously. The gallery on my studio web was make on this computer a few months ago. I have updated and used it to build a second set of galleries for a sister web for my landscape images. This makes absolutely no sense. I have not changed anything and the current 1.4.2 is working on my second mac with no issues. I really do not know what to do. As it stands, the software is totally useless. It creates the exact same files and folders that I just have deleted and trashed including the ELS. My studio software uses the same Adobe Air updater and with no issues.  I also have rebooted the drive after each deletion.

I hope someone can help me with this. I have two Mac machines running 10.9.5 both set up the same with the same apps. I have been using version 1.4.2. and am in the process of building my second web. 1.4.2 works fine on mac 1. On mac 2, I am getting a message pop up "JuiceboxBuilder-Pro wants to use your confidential information stored in "Adobe.APS" in your keychain. Do you want to allow access to this item?

Normally this is a pop up when you download new software from the web. This was happening on the original version. It is not happening on the other mac. I went ahead and tried upgrading to 1.4.3. It is still doing the same thing. It happens when I try to open the app, when I try to open a saved gallery, and I cannot even open new files without hitting "allow" repeatedly. including after choosing a gallery or file. It asks yet again. I cannot for the life of me find any difference in any settings on the two machines.
I desperately need to use this on machine two to get this web going. I am not sure what is going on. It does not appear to be related to the new update as it was happening on the older version.

And yes, Adobe Air is up to date. Any help would be most apreciated. I am DESPERATE at this point!

Thanks in advance!