SlickUpload is built around standard browser uploading, and thus require html file inputs.
If you want to change the look and feel of the control, you can use css skinning techniques as outlined here: http://www.quirksmode.org/dom/inputfile.html or here: http://shauninman.com/archive/2007/09/10/styling_file_inputs_with_css_and_the_dom. Simply apply those styles to the file inputs in the UploadHandler.aspx.