Unity Web Player install now button not working when JRE installed

We're running into an issue where left mouse clicking on the "install now!" button for the Unity Web Player no longer works. This appears to have started around the time of the 3.5 release and seems to occur on systems where a Java Runtime Environment is installed (JRE 6u31 in the tests I've ran using IE 8 under Windows 7). Our application uses a fairly standard HTML page for the Unity Web Player which accesses the UnityObject.js file at http://webplayer.unity3d.com/download_webplayer-3.x/3.0/uo/UnityObject.js.

As a workaround, it appears if you right mouse click on the "install now!" button and select "Open Link" you can get the installation to proceed; however, this isn't ideal. Another workaround is to uninstall the JRE which allows left mouse clicking on the button to work as expected.

Has anyone else seen this issue and is there a solution that allows left mouse clicking the "install now!" to work while a JRE is installed?