This tool works for all Instagram accounts, public and private, regardless of photo quality. Just search for the username you want and long tap on the circular image to view the full-size version. Alternatively, if the image is already large enough for the display, you can download it manually.
Aside from the app itself, there are also several websites and apps that allow users to enhance their profile pictures on Instagram. In order to download enlarged profile pictures from these websites, users must enter the users' username. The app then extracts the profile picture at its original size and saves it to the Apps folder.