The short story: I wrote a custom Gyazo client and server script that lets you name your screenshots, save as PNG or JPG (the original allows only PNG), and an option to save the file locally.
I decided to call it Gyazaux. You can find the source code for the client here and the server here. If you want an executable file, you can find it (along with the PHP script) in a zip file here.
Walf|EE 