I'm not familiar with Cubietruck A20 hardware but as far as I know, FEL is not something static that stays on once enabled even after a power-off or a reboot.
On each boot process, as long as you do not hold FEL button, Boot ROM (BROM) looks for a magic string (eGON) to find a proper medium. This magic string is searched in the header part of the storage mediums attached to MMC0, NAND, MMC2 and SPI flash, respectively. If no valid eGON header is found in any of these mentioned interfaces, FEL mode is executed as a fallback.
So, if your device it automatically jumps to the FEL mode, chances are FEL button may be somehow (i think it's unlikely) stuck or the header part of attached storage devices are either corrupt or cannot be read.