It's unclear to me what you mean by "If I configure one username for all outgoing mails...".
By default the Exchange 2010 installation in Windows Small Business Server 2011 will permit clients with LAN IP addresses to connect to port 25 of the Exchange Server, authenticate with either Windows Integrated authentication, or Basic Authentication if TLS was negotiated, and relay messages.
It sounds like you might be trying to send messages with "From:" addresses that correspond to Exchange recipients other than the one you're authenticating as. By default Exchange doesn't allow this and will return a "550 5.7.1 Client does not have permission to send as this sender" error. You can circumvent this functionality with the
ms-Exch-SMTP-Accept-Any-Sender permission, but you're compromising security when you do this (allowing users to forge "From:" on messages that will then be processed by Exchange before delivery, giving them an air of legitimacy). It would be best if you could configure multiple sets of credentials in Thunderbird (which I can't tell you about because I never could stomach using it).