How to choose a windowing function when performing a Chirp $\mathcal Z$-transform?
When we add the window function to the definition of the DFT we get

$$ X(k) = \sum_{n=0}^{N-1} w[n]x[n] e^{-j \omega n} $$
with $\omega_k= \frac{2\pi k}{N}, k\in \mathbb{Z} , $ $w[n]$  the window function and $x[n]$ the data. The multiplication of the data $x[n]$ with the window function $w[n]$ in the time domain corresponds to a convolution in the frequency domain.

The CZT and the FFT are basically just different ways to compute this sum.  Each algorithm has its advantages and disadvantages. The FFT is fast, the CZT is more flexible. If you use both algorithms  to compute the same coefficients $X(k)$ you get the same results. So, in this case, the window choice considerations should be the same for CZT and FFT.

So far I can answer only have of the original question: If the CZT is used to compute the same frequency bins as delivered by the FFT, there is no difference with respect to window selection.

Note1: Are you aware of the issues regarding frequency resolution of different DFT algorithms, as discussed e.g. in this thread ? The frequency resolution topic is the same for "CZT vs. FFT" as for "zero padding of FFT" or for "Goertzel vs. FFT".

Note2: This was the basic message of my original post.

If  the CZT is used to compute the DFT on a finer frequency resolution, things are more complicated. In this thread the effect on the DFT is discussed. To make it short, the CZT interpolates the DFT. This interpolation makes window selection more involved.

If (as intended by the OP) the CZT is used to interpolate the DFT at a finer frequency resolution, the underlying maths remain the same, however the different frequency resolution must be kept in mind.

This is illustrated in a very simple example. The first figure shows the absolute values of a rectangular window. The FFT (without zero padding or similar things) computes the DFT right at frequencies where the window is zero, here marked in orange. 


With finer frequency resolution this changes, as shown in the next figure.
 This may (as in the example) introduce additional leakage.