Windows Live One Care handles networks and adapters much like Vista does, and will need to be configured as such.
First, your will need to ensure Hamachi is allowed through your Firewall:
Open the Live One Care program.
Go to Change Settings.
Select the Firewall tab.
Here, ensure that Hamachi is allowed.
Next you will need to allow ping requests. This is done from the Ports and Protocols tab which can be reached through the above instructions. You need to enable ICMP v4 incoming requests.
Finally, you need to click on the Network Connections tab, and ensure the Hamachi network is set to Private/Work.