This occurs because of the Great Firewall of China.
The following content on the https pages (sign up and log in; are there others?) is blocked when accessed from mainland China without a VPN:
- OpenID functionality (openid.stackexchange.com)
- Facebook functionality
- Google account functionality (not strictly blocked, but heavily throttled, to the point of being effectively unusable)
Note that the "Welcome back $! You've been logged in. Click here to refresh the page." banner will function correctly even when not on a VPN, presumably because it uses the existing cookie. Obviously this won't help someone trying to make an account or log in for the first time.
Additionally, imgur is also blocked, so any images hosted there will not appear, though this is unrelated to the question at hand.
If you want to log in or sign up on SE from China, you need a VPN. Once you are logged in, you do not need a VPN to use the sites normally (non https pages), unless you log out.