Does Hamachi require NAT

Short version:
Hi my Internet Provider said i will need NAT to use hamachi, is i true?
Long version:
Me and my friend wanted to play minecraft so my friend created a Minecraft Server.
In hamachi network there are 5 devices:
- server
-me (connecting on ethernet cable to router)
- 3 of my friends
In 2nd network there are 2 devices:
server
my sister (using wifi)

I have direct connection (green dot) with my friends but the server is tunneled connection (blue dot).
My sister has blue dot on server.
Everyone else has green dot with server.

I worote a script that will restart hamachi if there is light blue dot next to a server, after about half hour of costant restarts server has a direct connection.
After contacting my IP they told me that I will need to buy NAT for my network.
I already did port forwarding and added hamachi as an exception in firewall.