I don't know if it was always nonsense, but I'm here to inform you that on Windows 7 x64 it is completely unnecessary to turn down secure mode, or find some elusive Privacy-setting to allow this site.
In Internet Options, beside your "Security" Tab, where you'd see "secure mode", should be a "Data Protection" Tab. There you see "Cookie Settings". As default it will be set to "Block all Cookies". It is completely sufficient to lower this setting exactly one step down, any more would also be unnecessary. Then you can keep Secure Mode enabled and Hamachi also connects (or let's you log in for that matter).
The only other thing I did, which I don't know if it had any significance or not, was going into my windows firewall settings and allow LogMeIn Hamachi (didn't need to enable "publicly"). I had to manually add LogMeIn Hamachi to the list though.
I hope this helps anyone on Windows 7 or above at the very least.