In this case the order doesn't really matter. You can do it either way. The order is only important when using the Convert Drive method, in which case it is always the firmware first and the EEPROM Utility afterwards.
Updating the boot code is not strictly necessary, so ticking this box is optional. Some recommend it, because it ensures that you always have a bootcode perfectly matched to the drive model (for what is worth), and as such the drive is as close to the original as possible. I guess that technically speaking updating the bootcode does carry a slightly higher risk than usual if you're unlucky enough for something to go very wrong during the flashing process, like a power cut occurring at the worse possible time during the flashing process, but the risk is rather negligible I would say, so you might as well tick the box. After all, LiteOn drives have proven over time to be very resilient and difficult to kill