Trying to verify the solution to a lambda calculus equation
On trying to verify this I first evaluated the L.H.S to be 
  
  $ GX $ = $ Y ( \lambda gx.gg ) $ $X$.
  
  Now $X$ being the argument will be picked up by the first bound
  variable  so .
  
  $L.H.S $ = $ Y ( \lambda x . XX) $


This is where you went wrong, and the reason is that you've got the association the wrong way round. $Y ( \lambda gx.gg ) X$ means $(Y ( \lambda gx.gg )) X$.

Taking that into account, the only evaluation step possible is to apply $Y$, giving $$( \lambda gx.gg )(Y ( \lambda gx.gg )) X$$ which is $( \lambda gx.gg )GX$, from which the desired conclusion is immediate.