The GNU Radio Live SDR image [1] does not have persistence, so you will not be able to permanently save anything to the USB drive. You could use a second USB drive or a network drive (Samba, SSHFS, NFS), and write to that. The Ettus Research Live SDR image [2] does have a way to install it that includes persistence, but that image is one year old, and uses old versions of UHD and GNU Radio, and will not work with some newer USRP hardware. I would suggest using the GNU Radio Live SDR image and using a second USB drive or network drive for storage.
There is work currently being done to add persistence to these Live SDR images. A new updated image could be ready in a month or two. Please feel free to check back with me directly.
[1]
https://gnuradio.org/redmine/projects/gnuradio/wiki/GNURadioLiveDVD[2]
http://files.ettus.com/liveusb/3.0/