firefox probably saves data somewhere under /home/tc - since you are using read-only media, you cannot save a backup.
You should be able to make a permanent home directory on a usb stick with the boot code "home=".
Since the usb stick location may change from sda1 to sdb1, etc, it is better to use a uuid to identify it, like the following example:
home=UUID="8eb3e1f8-c1ee-2bac-f66d-070f73960f62"
Use the following command to find the uuid of your usb stick: $ blkid -s UUID /dev/sda1