I think support to move a freedombox system to another drive is still missing.
Your easiest option may be doing a backup and then restoring it after configuring a fresh freedombox image that you could write onto the new drive as usual. The backup does not include the users for example.
Cubietruck is based on Allwinner A20 processor. So, here is what I would do this working.
Transfer FreedomBox image to SSD. If you are setting it up fresh, this is easy. While running from SD card, download the FreedomBox image and use the dd command as described in the manual page. If you want to transfer existing FreedomBox contents of SD card to SSD, then use any disk imaging tool to do this.
Prepare a minimal SD card, with u-boot on it. This is because, IIRC, the ROM code on the Allwinner A20 can’t boot from SATA. Do the following to an SD card with no data on it: