Opening a gallery (created in Lightroom or by any other method) in JuiceboxBuilder-Pro should not change the order of the images (unless you change the order yourself by dragging and dropping thumbnails on the 'Images' tab or by using the 'Images -> Sort' options from the drop-down menu at the top).
JuiceboxBuilder-Pro should just read the list of images from the 'config.xml' file and initially display them in the order that they are listed in.

Any additional images you add to a gallery in JuiceboxBuilder-Pro are added to the end of the gallery. They are not automatically put into alphabetical (or any other) order. However, once you have added all the images for your gallery, you can re-order them using the 'Images -> Sort' options from the drop-down menu at the top.

What version of JuiceboxBuilder-Pro are you using ('Help -> About JuiceboxBuilder-Pro...')?
The current version is v1.4.0. If you are not already using v1.4.0, then please try upgrading following the instructions here.

If you are already using v1.4.0, then it certainly sounds like something is amiss. I cannot replicate the problem. Dragging and dropping images into a new order works fine for me, as do the 'Images -> Sort' options.

Try uninstalling and reinstalling JuiceboxBuilder-Pro to see if this helps.

Does this happen with all galleries (try creating a new sample gallery with a few images in JuiceboxBuilder-Pro) or just certain ones?

Thank you for the additional information.
I have been able to replicate the problem you describe and have logged a bug report with the developers.
It should hopefully be fixed in the next version.
In the meantime, depending on how you are trying to sort the images, I notice that sometimes, the action you perform is executed OK but requires a click on the 'Images' tab label (or a click on 'Customize' and then back to 'Images') to refresh the page (and display the images in the new order).