my phone: Verizon Galaxy S III i535 (not sure if this would work on the S III from other providers, I imagine it would though)
To disable the update notifications and stop it from auto downloading:
1) root the phone using towelroot xda towelroot thread and towelroot download
2) install a file browser with root capabilities; I have always used Root Browser by JRummy
3) using the file browser, navigate to /system/app and rename two files (provide root access when it prompts for it):
FWupgrade.apk to FWupgrade.bak
SDM.apk to SDM.bak
--- that has fixed the "we want you to take this crappy update" notifications for me. now if your phone has already downloaded the firmware (400mb file) this is how to remove it:
4) navigate to /cache/fota/ and delete "update.zip"
That has fixed the annoying issue of it wanting to update every day and freed up some space too. I hope this helps someone else out!