First, let's find the expression for the $n^{th}$ term of the sequence. We know that:

$$S_n = a_1 + a_2 + \dots + a_n = n^2$$

To find the $n^{th}$ term, we can use the difference of consecutive sums:

$$a_n = S_n - S_{n-1} = n^2 - (n-1)^2 = n^2 - (n^2 - 2n + 1) = 2n - 1$$

Now, we can find the required terms:

$$\begin{aligned} a_4 &= 2(4) - 1 = 7 \\ a_5 &= 2(5) - 1 = 9 \\ a_6 &= 2(6) - 1 = 11 \\ a_7 &= 2(7) - 1 = 13 \\ \end{aligned}$$

Finally, we find the sum of these terms:

$$a_4 + a_5 + a_6 + a_7 = 7 + 9 + 11 + 13 = \boxed{40}$$