Mike,
I'm afraid I'm running out of ideas as I've never seen that behavior before. In Fiddler I can see that there is a 302 redirection to the login control as soon as the Gallery module attempts to load DesktopModules/Gallery/GalleryPage.aspx which is a wrapper around various views when pop-ups are enabled in the Gallery configuration. GalleryPage does check for a number of potential security violations and if any are found there will be an intentional redirection to the DotNetNuke access denied url which will by default bring up the login control. Here are the checks - perhaps one of they might bring to mind something that is different about your site:
1. The tabid value (90) contained in the GalleryPage.aspx querystring must be the same as the currently active page. It looks to be correct and it does not appear that the site is multi-language which might be a factor.
2. The mid value (432) contained in the GalleryPage.aspx querystring must be that of a Gallery module on the currently active page. That looks to be correct but as part of this test the DesktopModule.ModuleName property is assumed to be "DNN_Gallery". Please check in the Host-->Extensions configuration for the Gallery module that this name was not changed to something else.
3. The user must have view permission on both the gallery page and the module - it appears that this is the case for all users including unauthenticated users.
4. The ctl value (viewer) in the GalleryPage.aspx querystring must be one of 5 values of which "viewer" or "Viewer" is one.
Please check the Admin-->Event Viewer to see if any related errors are logged when clicking on a Gallery thumbnail.
You might try turning off the Enable Popups in the Gallery configuration to see if the need to login is removed and if you can tolerate having the image viewer displayed in place.
Another possibility would be to consider upgrading the site to DNN 5.6.2 (minimum but 5.6.7 recommended) and upgrading Gallery to 4.4.0, although I don't see that any issues such as yours were fixed in Gallery 4.4.0. I do recall, however, that DNN 5.5.0 introduced an issue with custom aspx pages inheriting from PageBase (such as GalleryPage.aspx) throwing an exception. It was fixed in DNN 5.5.1. If you do upgrade, be sure to follow the usual good practices of backing up both the file system and database prior to the backup. Also, Gallery 4.4.0 made extensive changes to the html markup of all controls and to the css style sheets of the Gallery themes for better XHTML compliancy. Any customizations to style sheets that you have made will have to be re-done for Gallery 4.4.0.
Let me know if I may be of further assistance.