I had a few issues with auto log in on my home laptop, and I suspect it was something to do with my antivirus software which was trying to do an automatic update.
Unfortunately I don't know for sure what was going on, but it's working for me now,kinda...
If I go to the beior.org homepage, I am automatically logged in. If however I go directly to the Board Index page, I don't get auto logged on - easily rectified by clicking on the homepage and then back to the index though.
I've found my work PC will stay logged in for a good while if I log in through the homepage, but it doesn't last til the next day.