Sketchup uses the default browser of your system; Internet Explorer on Windows and Safari on the Mac. You can set other (like Firefox) to be the default browser for Internet but not for Sketchup - so it will go on using the originaldefault browser.
Anyway, this shouldn't prevent you from uploading a model provided you are logged in via SU as well. But when you try to upload something, it should tell you to log in (from SU) and when you have done so, it should let you upload the model.
So either with the G button or on the upload page of SU, can you log in or is there a problem with logging in at all?