Yes, indeed you can! Click on the "Image button" beside the Icon title and a window of available icons will pop up. Instead of choosing the icon from the library, click on Upload and select your own icon from your computer.
Your image must either be a .jpeg or .png format.
Ultimately, icons need to be 512x512px in size. If you want to keep your text as a rectangle, that's fine, but you will have to create a square image with the text 'in the middle' so that the apps will have a square icon to display.