The browser works because you are essentially reverse proxying through the UAG to the connection servers web portal. When you establish a connection to the backend resource (desktop/RDSH) through the portal you are connecting from the connection server.

The error you mention has me suspecting that you may not have the appropriate ports open from the UAG to the resources (desktops / RDSH) on the backend. If you review the logs on the UAG should indicate the problem.

Since you can authenticate using the client and connect successfully to the HTML portal it sounds like your HAProxy is setup correctly.