Linear Convolution in DSP and Hann window
This is an insightful question, one that I remember pondering in the 1980s when I first learned of the Overlap-Add (OLA) and Overlap-Save (OLS, sometimes called Overlap-Scrap) methods of "fast convolution".

It turns out that for strict LTI filtering, that you need not window with a Hann, but you could if you wanted to, and for a general frequency-domain modification algorithm, you may want to window with Hann anyway.

Let $w[n]$ be your window and have non-zero length $L-1$ which is less than the length of the FFT, $N$.  If it's Hann, the symmetrical definition (with $L$ even) is:

$$ w[n] = \begin{cases}
\tfrac12 \cos\left(\tfrac{2\pi}{L}n \right) + \tfrac12 \qquad &amp; \text{if } |n| &lt;  \tfrac{L}{2} \\
\\
0 \qquad &amp; \text{if } |n| \ge \tfrac{L}{2} \\
\end{cases} $$

You can see that 

$$ \sum\limits_{m=-\infty}^{\infty} w[n-m\tfrac{L}{2}] = 1 \qquad \forall n \in \mathbb{Z} \tag{1}$$

then

$$\begin{align}
 x[n] &amp;= x[n] \sum\limits_{m=-\infty}^{\infty} w[n-m\tfrac{L}{2}] \\
\\
&amp;= \sum\limits_{m=-\infty}^{\infty} x[n] w[n-m\tfrac{L}{2}] \\
\\
&amp;= \sum\limits_{m=-\infty}^{\infty} x_m[n-m\tfrac{L}{2}]\\
\end{align}$$

where $$ x_m[n] \triangleq x[n+m\tfrac{L}{2}] w[n] $$ is a "grain" of $x[n]$ slid over to be centered at $n=0$.

You can see you can take each segment or grain, $x_m[n-m\tfrac{L}{2}]$, slide it over to 0 to be $x_m[n]$, pass that through your filter (however it's done, with an FFT or not) to get $y_m[n]$ (which is not normally as short as $L-1$, but must be no longer than the FFT size, $N$), which you slide back to $y_m[n-m\tfrac{L}{2}]$ and add up each term to get $y[n]$:

$$ y[n] = \sum\limits_{m=-\infty}^{\infty} y_m[n-m\tfrac{L}{2}] $$

But this works whether the segments overlap or not as long as Eq. $(1)$ is valid.  You can make this work with a rectangular window.

But, for a rectangular window, the hop length (which was $\tfrac{L}{2}$ for the overlapping Hann) will be the same as the non-zero window length for a rectangular window (which was $L-1$ for the Hann).  So overlap-add will be more efficient with a rectangular window.  But you might want to do other frequency-domain things (like a phase vocoder) so I might recommend using the Hann window anyway for OLA.  There is something elegant about it.