You do not need a flash programmer. Programming is done by the bootloader program that is already set up in your chip.
What you need is a linux Bootloader host. I don't know if there is one. Enter "linux bootloader host" into the keyword search field, there are some matches.
Seems like this might be a good start:
C:\Program Files (x86)\Cypress\PSoC Creator\3.3\PSoC Creator\cybootloaderutils
From the readme.txt:
This directory contains the source files for the Cypress-provided Bootloader Host Tool, which is used test out the bootloader running on a PSoC chip. For more information about this tool, refer to the Bootloader Host Tool documentation (F1 from the application). For more information about the source code in this directory, refer to the Bootloader Component Datasheet."