I'm going to post this as a public service because it gave me trouble for days. If you want to enroll in the Windows 10 Extended Update Program (EUP) to get another year of Windows 10 security updates, I figured out the following:
You need to be logged into a Microsoft account in Windows. (You WILL NOT have to stay logged in indefinitely, but you have to be logged in to do the enrollment thing.)
Before trying to enroll in the EUP, you need to go to Settings > Privacy > Diagnostics & feedback and make sure "Tailored experiences" is turned ON. If it's not, the EUP enrollment will fail and it won't tell you why.
The EUP enrollment link is under Windows Updates. They took their time rolling it out, but if you got the September update, it should appear there now if it hadn't before.
If you try to enroll and the window just disappears without telling you if it's succeeded or failed, you will need to restart Windows or it'll keep doing the same thing.
Once you've successfully enrolled, you can turn OFF "Tailored experiences" and log out of your Microsoft account. (Microsoft is obviously trying to trick people into leaving "Tailored experiences" on and staying logged in order to maximize their data harvesting, but you don't have to, and you shouldn't.)