Can I view my MSSQL connection string from within SQL Server Management Studio?

You can. From Stack Exchange:

select
    'data source=' + @@servername +
    ';initial catalog=' + db_name() +
    case type_desc
        when 'WINDOWS_LOGIN' 
            then ';trusted_connection=true'
        else
            ';user id=' + suser_name()
    end
from sys.server_principals
where name = suser_name()