NeoPixel Ring Usage Without Arduino
No, the settings for the colors are stored in RAM on the individual devices so they need to be reprogrammed every time they are powered on.

On the other hand, it is possible to use a much smaller device such as a surface-mount ATtiny85 to program them, and light_ws2812 runs on said device just fine. And once they have been programmed it is possible to put the MCU in deep sleep so that it consumes almost no power.