Charles, thanks for this solution. I had a similar issue in Visual Studio 2012. What worked for me in View->Code was changing:

DTS:ColumnNamesInFirstDataRow="True" to DTS:ColumnNamesInFirstDataRow="False"

Going back to the Connection Manager, the box was unchecked, but the column names were still correct. I ran the package and all of my data rows loaded correctly. When I go back to View->Code, the DTS:ColumnNamesInFirstDataRow field is no longer visible as it would be if I left the box unchecked in the first place. This is perfect so thanks for the tip and thanks DB_Newbie for asking my exact question.