Exactly!
I never had problem until I bought PC with Win 8.1 or Win 10. Computers with SO: XP, Vista and Seven, work perfectly. I have your same doubt: I gave for sure that if a PC is turned off with Hamachi online, it would be online also at restart (a sort of last state), but I'm not sure this is a normal perfomance of the app. It was so, until now, but probably there is not a real control in the software . Same PC sometime starts up with hamachi online, sometime in offline state. It's randomic, or it appears so, anyway!
So I also think hamachi have to insert a sort of setting to ensure the online state at the startup of the PC.
Anyway I'm mailing with customer support, trying to explain the problem and when it occurs. Anyway I think that this thread should be sufficent for them to understand that a problem exists and probably also where to look for, in their software.