Most regular web hosts that provide 'plain vanilla' web space that you upload files and folders to via FTP also offer PHP but it sounds like your web host might be more of a web template site (which might explain why they do not offer PHP).
The two PHP files within the Juicebox-Pro 'jbcore' folder ('juicebox.php' and 'pswd.php') are required only for gallery password protection and direct download functionality.
As long as you do not use either of these features (and, therefore, do not set either usePassword="TRUE" or "showDownloadButton="TRUE"), then your gallery will function fine without the PHP files being present within the 'jbcore' folder.
The password pop-up box on your site is not Juicebox-Pro's own password protection.
To see Juicebox-Pro's own password protection, please take a look at this demo gallery.
The password pop-up box on your site looks like directory password protection at server-level (having been put in place via either a '.htpasswd' file on your web server or your web hosting account's online Control Panel).
I notice that your entire '/dav/' directory (a parent directory of your gallery) is password protected in this way. If you go to https://mithril.ie/dav/, then you'll see the password protection. Please ask your web host why this directory is password protected and how to remove this protection. Once the password protection has been removed from your '/dav/' directory (and all its subdirectories), then your gallery should hopefully display and function fine.
I hope this helps.
Please let me know how you get on and if I can be of any further assistance. Thank you.