Well I use my router with ExtRoot so I can just give you an overview on how to do it. I am considering that you have a default config for your router.
First you should identify if your USB is plugged in properly and for this try ls /dev/ and you should see all the partitions. Also make sure you have the correct filesystem installed for example Ext3. Then mount your usb partition with mount /dev/sda1 /mnt. sda1 is the first partition of USB. Edit the /etc/config/fstab to make sure everything runs on boot.
After that you need to copy your current /overlay to usb partition and for this use this command. tar -C /overlay -cvf - . | tar -C /mnt -xf - and once you are done then just umount /mnt and reboot to see the changes.
Edit: Please post the output of cat /etc/config/fstab here