Would I need to swap my original ecotec gearbox if I convert the engine into L67?

No, however, you would definitely need to use the L67 flex plate, otherwise, the engine will be out of balance. If you currently have a manual transmission, then you'll need to get your flywheel mirror balanced to suit the original L67 flex plate. The main reason for this is that these engines, including ecotecs, are externally balanced engines from the factory.