I do not think one can push the user credentials from the browser to the database (and does it makes sense ? I think not)
But if you want to use the credentials of the user running Tomcat to connect to SQL Server then you can use Microsoft's JDBC Driver. Just build your JDBC URL like this:
jdbc:sqlserver://localhost;integratedSecurity=true;
And copy the appropriate DLL to Tomcat's bin directory (sqljdbc_auth.dll provided with the driver)
MSDN > Connecting to SQL Server with the JDBC Driver > Building the Connection URL