Consider the following expression:
$$x^4 + y^4 + (x + y)^4 = 2x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + 2y^4,$$
thus
$$2(x^4 + y^4 + (x+y)^4) = 4(x^4 + 2x^3y + 3x^2y^2 + 2xy^3 + y^4) = (2(x^2 + xy + y^2))^2.$$
Hence for $x = 20$, $y = 19$ the following holds:

$$n^2 = (2(x^2 + xy + y^2))^2 \text{ or } n = 2(20^2 + 20 \cdot 19 + 19^2) = 2282.$$