Writing to an USB stick may be a problem since the PIC is an USB-Slave, just as the USB stick. Writing to a SD-card or similar should be possible with PBP but there aren't any ready made "commands" to do so, you need to write the routines yourself.

If you want to take an easier aproach you might have a look at modules like DOS-on-chip or the MicroDrive. They interface to your PIC via RS232 or SPI and handles all the reading/writing to the SD-card. There are other similar products but their names escape me at the moment.