jmc

Suppose $xy$ is negative. If we flip the sign of $y,$ then we flip the sign of $xy,$ which makes it positive. This increases by the value of $x^6+y^6+xy,$ so if $x^6+y^6+xy$ is minimized, then $xy$ must be positive. We may assume that both $x$ and $y$ are positive.

By AM-GM, $$\frac{x^6+y^6+27+27+27+27}{6}\ge \sqrt[6]{(x^6)(y^6)(27^4)}=9xy,$$which simplifies to $x^6+y^6-54xy\ge -108.$

Equality occurs when $x^6=y^6=27,$ which leads to $x=y=\sqrt{3}.$ Therefore, the minimum value is $\boxed{-108}.$