Enemy is losing 2 health instead of 1
I think you have twice as many components as you need here. While I'm generally in favour of small single-purpose classes, "have a health value" is so tightly linked to "modify health ...
Unable to access an instanced health system
You have a variable private HealthButtons _healthButtons; in your GameHandler script, but you are not setting it to anything in code. And you wouldn't be able to ...
How do I send data from client to server using LiteNetLib
You can use client.SendUnconnectedMessage(packet,peer.EndPoint) to send message to a fix endpoint - in your case the endpoint is the server.
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