Answer: 44

Small Hint:

Factor the expression before raising it to the $n$th power

Big Hint:

Count the distinct choices of the exponents of $x$ and $y$

Solution:

The base factors as $$\begin{gathered}xy-3x+7y-21\\=(x+7)(y-3).\end{gathered}$$ Hence its $n$th power is $(x+7)^n(y-3)^n.$ Each exponent of $x$ from $0$ through $n$ can occur with each exponent of $y$ from $0$ through $n,$ and every resulting coefficient is nonzero. Thus there are $(n+1)^2$ terms. Since $44^2<1996\leq45^2,$ the least possible $n+1$ is $45,$ so $n=44.$