I really prefer the traditional square avatars! When I've been doing forums for over a decade, I've always seen them square shaped, *opinion alert* and are more traditional than those round ones that those fancy paid forums always use! *end opinion*
I prefer Square Avatars because I can show more of the background picture, and text on the avatar instead of being limited to a circle which has less space than a square which is the same height and width.
A square also shows the entire picture instead of a circle where the corners are missing. I can also add a border or frame design to surround my avatar to make it look like a old fashion Polaroid or postcard picture.
A square avatar can also be used to put a square QR barcode if I want to use my avatar image to share a website link or other information with a square QR code on my avatar.