Hello,
When you have a free hamachi it runs as a Local service so it won't automatically connect when you reboot your computer. Now a Paid hamachi version will run the hamachi service as a System service so it will allow you to automatically connect to your networks uppon reboot.
If for some reason the service has trouble starting upon restart the usual fix is to open your services.msc > right click the Hamachi Tunneling service and then choose it for Automatic start with delay.
Best,
Vitor