I would recommend just installing the Juicebox Lightroom Plugin afresh on your new computer.
For each licensed copy, you can install and use JuiceboxBuilder-Pro (and the Pro plugins) on up to 3 different computers, as noted in this FAQ:
Can I install on multiple computers?

If you still have your Juicebox-Pro zip file available, you can copy it to a USB flash drive and transfer it to your new computer.
If your zip file is not the current version (v1.5.1), then I would take this opportunity to download and install the latest version on your new computer.
If necessary, you can download Juicebox-Pro v1.5.1 using your original download link. Upgrades are free within the same version number and the download link always points towards the latest version rather than the version you purchased.
If you cannot find your original download link or if it has expired, then please let me know and I can send you a new one.

Once you transfer the Juicebox-Pro zip file to your new computer (or download it directly onto your new computer), just follow the installation instructions on the plugin's support page here.

Hopefully these notes will help.
However, if you get stuck with anything, just let me know how far into the process you get and what seems to be the problem and I'll do my best to help you out further.