I contacted my server (Lunarpages) about this problem – here is their reply:
The issue is due to the recent WordPress Update that requires connection to www.remarpro.com to only use the latest version of OpenSSL. Unfortunately, you are on CentOS 4 server and we cannot upgrade OpenSSL on CentOS 4 machine as the latest OpenSSL only support CentOS 6.
However, we can move your account to another server with CentOS 6 and the latest OpenSSL if you wish.
I am going to have them move my account and will let you know if I can then update to 4.4.2