You should be able to directly replace your Picasa User Id and Picasa Album Name in the WP-Juicebox interface with your Google User Id and Google Album Id/Name respectively.

How to find your Google User Id:
You can find your Google User Id by logging in to Google+ and clicking the 'Google+ Profile' link above the 'My Account' button (after clicking your custom image at the top right). Your Google User Id will be the 21 digit number in the URL (in your browser's address bar).

How to find your Google Album Id/Name:
Visit this URL in your browser and you'll get an RSS feed of your albums: https://picasaweb.google.com/data/feed/api/user/<User Id>
If you then hover over (or click on) an album, you'll see a 19 digit number (or possibly a name if it's an older album) at the end of the URL. This is the Album Id/Name which you can use in the WP-Juicebox interface.

Make sure album is public:
Visit this URL in your browser: https://picasaweb.google.com/<User Id>/<Album Id>
In the right-hand sidebar next to the thumbnail representing the album, make sure that the visibility setting is 'Public on the web'. If the visibility is set to any other value (e.g. 'Limited' or 'Only you'), the click the 'edit' link and select 'Public on the web'.

Hopefully this will help.

If you continue to experience difficulties, please let me know your Picasa User Name and Picasa Album Name and I'll try to help further. Thank you.

If I go to https://picasaweb.google.com/data/feed/ … 8088651929 I am presented with an RSS feed of all your albums so 117363164378088651929 is certainly a valid Google User Id.
However, the Google Album Id 6315232953461261745 (from the URL in your last post) does not appear to be a valid Google Album Id for that account. That Google Album Id does not appear anywhere within the RSS feed data.

I do not know which album you are trying to display but if you use a Google Album Id from the links in the RSS feed page, the corresponding gallery should display fine.

This is certainly a problem related directly to Google+ (rather than Juicebox) and you might have better luck posting in a Google+ support forum.
However, it looks like Google are slowly decommissioning Picasa Web functionality.
As a test, I just created a new gallery under Google+ and although it appears in my Picasa RSS feed (so it appears to be public), its Google Album Id does not seems to be working in the Picasa Web Album API (and, consequently, the album cannot be displayed by WP-Juicbox).
Only Google really know what functionality they provide and there seems to be little definitive information about this subject on the web. (If you do a web search for 'make google photos album public' you'll find a few forum threads relating to this subject but it's as much information as I've found.)
The best information available is perhaps experimental evidence gained from your own findings and my own tests.
It looks like it is no longer possible to change the accessibility (private vs public) of a Picasa Web Album (although I cannot be certain about this, hence my suggestion to post your query in a Google+ forum) or to use the Picasa Web Album API in conjunction with a new (post-Picasa) Google+ Album Id.
Unfortunately, there is nothing that we can do to change this and, as far as I am aware, there is no workaround (other than to use a different platform to store and organise your images such as Flickr).