No, it will not work because the tool that does the firmware update is Windows only. I wouldn't recommend using Windows emulation (BootCamp or Paralells Desktop) or even Wine Bottler to try to run it.
Flashing the firmware in a wrong way might render the device inoperable with low chances of recovery. Always use the indicated operating system when you're install firmware updates for a device.
Access the following website to learn about the update process for he C7042 device.