hmm...

is the chess.com logo supposed to be a red square with a BF in it?

like the one on my browser

what do you mean? 

on the tab the logo was a red square with a BF in it instead of a green pawn...

its better now, but before...

Groups and members can customize their homepages. If you were visiting someone's homepage you might've seen something differing from the original logo. Here's an example: http://www.chess.com/groups/home/the-brain-strainers-of-chesscom

not the chess.com thing with the pawn and play learn share, but the logo beside the internet address...

meh, its better now

When I stay on chess.com for a long time then go to another site, that logo sometimes sticks.  Maybe the logo of another website that you had just visited was still up there.