An older post said to login to your account within itunes; go to account information; then disable icloud and ping. Since I'm also on 10.6.8 there is no "disable iCloud" option. So this is what I did (and it worked) - from the Account page within iTunes Store (must be logged in):
1. Ping was already disabled/off
2. In the iTunes match section, I clicked "Learn More" then clicked "No thanks". Effectively turning off Match (it gets removed from the sidebar).
3. Under "Settings" I turned off "Recommendations and Ping Postings".
I don't know which step - #2 or #3 actually made that login prompt no longer appear, but it doesn't so I'm happy.
Also, I had already disabled "Download updates automatically" in system preferences (and that didn't fix the iTunes problem). Note that I had that disabled that before I updated to 10.6.8 and the 10.6.8 re-enabled it. Not related, just mentioning in case that box needs to be unchecked too.
Other things you can try are in this Reset Warnings tip in this iTunes Windows thread (I didn't try it since the above worked for me):
https://discussions.apple.com/thread/3524167?answerId=18589065022#18589065022