We have the same on our client computers. Hamachi poweroff himself. Only with a Power-on click it logged on again. I also don't get it why there is a power on/off button if you run hamachi by services. The services run, but a client can overrule by clicking power-/loggoff button. Bye bye connection.
We also asked support to change this or make a command option to auto poweron hamachi interface we can builld in a script. But no support from here, and we also have a paid account with 999 clients.