Although I'd hate to be the one saying "you can't", but unfortunately, I have to. As I've been an experienced Windows user, I haven't seen any programs that can modify the Start menu. I have seen a few designed specifically for Windows 8.1, but they only really change the look and animations within the Start menu. If you were able to remove the square holding the program's icon, it'd probably be hard to see where to move the programs inside of the Start layout.
But hey, it doesn't have to end there! If you are familiar with C++ or XAML, you could probably program something to get where you're trying to get to, which is your desired customization.
I'd also try searching the internet tirelessly, as it always works, just to find what you're looking for. It's really just a 50/50 chance, but you may end up finding it if you look hard enough!
Be on a lookout for the new Start layout, too, here:
https://www.theverge.com/2019/7/24/20708355/microsoft-windows-10-accidental-release-windows-insiders-start-menu
https://www.windowscentral.com/hands-new-start-menu-windows-10-build-18947