I would suggest astartingpoint for setting up a proof might be (assuming that you're implying that X and Y can be correlated):

Let Z, E1, and E2 be independent standard normal random variables. Let X and Y be linear combinations of these variables as follows:

X = r*Z + Sqrt[1-r^2]*E1

and

Y= r*Z + Sqrt[1-r^2]*E2.

where 0<r<1.

Now, determine the variance of X*Y by taking expectations (and using the definition of variance) of the product of the expansion of the right-hand sides of X and Y. Hint: the correlation between X and Y will be r^2.

Generalize your result.