It doesn't sound like you're doing anything wrong.
I've just tried to replicate the problem you describe but everything seems to be working as expected for myself.
With a fresh Showkase v1.3.6 site, I created a Juicebox gallery page using default settings.
I then went to 'Site -> Customize Viewers', selected all image sizes in the 'Multi-size Images' section and clicked 'Update' and 'Publish'. The gallery was displayed in the browser and small and large images were created in the gallery folder's 'images/small/' and 'images/large/' folders.
My test above suggests that the problem may lie somewhere within your own setup rather than with Showkase functionality (which seems to be working OK for myself).
I notice that your web site uses Showkase v1.3.2.
Please try upgrading to the latest version of Showkase (v1.3.6) following the upgrading instructions here.
If the problem is related to a PHP memory or timeout issue (which is possible if you have many gallery pages and images which Showkase has to process), then upgrading may help as Showkase v1.3.4 introduced the following changes:
(Please see the Version History for a full list of changes between versions.)
Hopefully upgrading Showkase will help. Otherwise, if you have access to the PHP settings for your web hosting account, please check the PHP MEMORY_LIMIT and MAX_EXECUTION_TIME values and try increasing them. Your web host should be able to help you out with this if necessary.
If you continue to experience difficulties, then it might also be worthwhile running the Showkase Server Compatibility Test to see if any tests fail.
You may have already run the test prior to installing Showkase but if your web server has changed anything recently, then running the Showkase Server Compatibility Test again may help in diagnosing the cause of the problem.