If you have a flash drive with you, your flash drive should be writable! :-)
Even though those machines might have a writable area, do consider the fact that they might have set some tool to auto-clean those areas every day! Something that clears all temporary files, for example. Or all downloads and other stuff. You might find some writable space on that disk, only to discover it has been wiped again the next day.
(My employer has set up a network disk where we can store stuff for the rest of the week. Every weekend, an automated process just deletes all content from this disk so it can be used for a clean start next week. Perfect for sharing files with colleagues while also making sure the file system doesn't clog up with all kinds of garbage...)
Furthermore, you should check their policies about this! They might not want you to use any custom executables on their system! What if your FireFox copy is infected with something nasty? You could be held responsible for any damage if they ever catch you using FireFox on their systems. My advise: if you want to use FireFox, bring your own laptop... (And use their network, if possible.)