Using:

tellraw @p {text:"Line1\nLine2 <-- click",color:green,clickEvent:{action:run_command,value:"/tellraw @p {text:'Line1\nLine2',color:red}"}}

at a command block, I get this message in the chat:

Line1
Line2 <-- click

After clicking on it I get:

Line1 Line2

I think, to use the "\n" in "Line1\nLine2" I should get:

Line1
Line2