Solving a Linear Mean Square Estimation the Easy Way
So in your case doesn't the relation $x = n+y$ help ?

I mean, assuming your derivation for the mean square estimtor is right, then to compute $E\{xn\}$ you would look for $E\{ (y+n)n\}$ and using properties of $x$ and $n$ you would get

$$E\{xn\} = E\{(y+n)n\} = E\{yn\} + E\{n^2\} = 0.5 + 1 = 1.5 $$