squashfs and axfs are read-only file system. There is no R/W option.
squashfs is a compressed file system, so the flash images are smaller and reading out blocks are quick, but all code blocks need to be copied to system RAM first. Being Read-Only helps make it fast.
AXFS is a XIP file system, and also being read-only makes it fast. But, unlike squashfs, file blocks remain and execute in place from Flash which reduced RAM usage and shortens (eliminates) page fault delays.
If you want a R/W file system in QSPI, then you can't use a XIP kernel. You need to use a RAM based kernel (uImage) and you can only use a single SPI flash (not dual).
If you look at file "doc/testing/spibsc-qspi.txt" in the BSP, it will explain how to create and boot from a JFFS2 file system using a single QSPI and that file system will be R/W. You can't use dual SPI flash in Linux because the Linux MTD subsystem has no idea how to deal with dual SPI flash addressing and such, but performance wise, it wouldn't make much of a difference anyway.
Chris