One possible proof is by induction. First, we use Euler's Theorem to simplify the statement:
$$a^{p^k + q^k} \equiv a^{n^k + 1} \pmod{n} ~ ~ ~ \Longleftrightarrow ~ ~ ~ p^k + q^k \equiv n^k + 1 \pmod{\varphi{(n)}}$$
Now, we show that the base case $k = 1$ is true.
$$p + q \equiv n + 1 \pmod{\varphi{(n)}}$$
And as $\varphi{(n)} = (p - 1)(q - 1)$, this is true (add $\varphi{(n)}$ to the left hand side).
Now, assume the statement holds for all exponents between $1$ and $k$ inclusive. Then we know that:
$$p^k + q^k \equiv n^k + 1 \pmod{\varphi{(n)}}$$
$$p + q \equiv n + 1 \pmod{\varphi{(n)}}$$
And so we obtain:
$$(p^k + q^k)(p + q) \equiv (n^k + 1)(n + 1) \pmod{\varphi{(n)}}$$
$$p^{k + 1} + q^{k + 1} + q p^k + p q^k \equiv n^{k + 1} + 1 + n^k + n \pmod{\varphi{(n)}}$$
So we see that if the statement is to hold true for $k + 1$, we just need to show that:
$$q p^k + p q^k \equiv n^k + n \pmod{\varphi{(n)}}$$
Divide this through by $n$, which is valid as $\gcd{(n, \varphi{(n)})} = 1$, thus:
$$p^{k - 1} + q^{k - 1} \equiv n^{k - 1} + 1 \pmod{\varphi{(n)}}$$
Which is the statement for $k - 1$, true by assumption! So it also holds for $k + 1$, and by induction holds for all $k > 0$.
Now consider the case $k < 0$. For this step, note that:
$$\left ( p^k + q^k \right ) \cdot n^{-1} \equiv \left ( n^k + 1 \right ) \cdot n^{-1} \pmod{\varphi{(n)}} ~ ~ ~ \Longleftrightarrow ~ ~ ~ q^{-k} p^0 + p^{-k} q^0 \equiv n^0 + n^{-k} \pmod{\varphi{(n)}}$$
$$\therefore ~ ~ p^{-k} + q^{-k} \equiv n^{-k} + 1 \pmod{\varphi{(n)}}$$
Showing that if the statement holds for $k$, then it also holds for $-k$. Finally, show the trivial case $k = 0$.
$$\blacksquare$$
Interestingly, this has a connection to the pairs $(x, y)$ which satisfy the following equivalence:
$$x^k + y^k \equiv n^k + 1 \pmod{\varphi{(n)}} ~ ~ ~ \Longleftrightarrow ~ ~ ~ \left ( x + y \right )^k \equiv \left ( n + 1 \right )^k \pmod{\varphi{(n)}} ~ ~ ~ ~ ~ \text{for all} ~ ~ k \in \mathbb{Z}$$
For which any solution $(x, y)$ must be some linear combination of $p$, $q$ and $n - 1$.
Note that the exponent space is not really $\mathbb{Z}$ but $\mathbb{Z} / \varphi{(\varphi{(n)})} \mathbb{Z}$.