Hi and welcome,
Have you tried swapping the red with the green or blue Arduino output connection to the FETs, and swapping the connections from the FETs to the strip? If red remains steady and blue/green still flicker, this would point to a hardware rather than software problem. (My suspicion is a hardware problem, since red LEDs generally have a lower forward voltage than blue or green. Are they logic-level FETs?)
Please post links to the data sheets or whatever info you have on the strip and the FETs, and a schematic diagram (this can be hand drawn and scanned/photographed if you like).
Paul