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Does anyone know if it's possible to remove my username from the Finder bar in the top right corner?
Do you mean the Menu Bar, at the top of your screen? When you click on it, does it give you an option for Account Preferences? If so, click that and go to Login Options and uncheck "Show fast user switching menu"

I'm not on ML, so you may be talking about something else.
 
Both, the option's pretty much the same in ML.

Can be enabled or disabled, and has the option of Full Name, Short Name or Icon when enabled.

I always leave it off now.
 
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