There is a very common equivalent: pourquoi pas ?

"Pourquoi pas ?" is indeed very common as a question for "why not?"
If you want to build a sentence out of it, you can say Pourquoi ne pas ____ ?, which stands for why not ____?

Why not going to Paris instead?

Pourquoi ne pas aller à Paris à la place ?

For queries like this one may consult the webpage linguee.fr.
In your case

https://www.linguee.fr/anglais-francais/traduction/why+not.html

shows how to interpret why not depending on the context. As the others have already pointed out it would mostly be pourquoi (ne) pas.

As a further example consider the case one wishes to find the interpretation of tag questions in French (…, isn’t it?), that is the kind of question which seeks confirmation of a statement just made.

Le voilà (typing isn’t it? in the search area):

https://www.linguee.fr/francais-anglais/search?source=auto&query=isn%27+t+it%3F

EDIT

One sees the common (mostly formal nonetheless; see the comments accompanied my response and the reference therein and the links below) n’est-ce pas ?

https://www.quora.com/What-does-Nest-ce-pas-mean-in-French-and-is-it-commonly-used-If-so-how

https://www.thoughtco.com/nest-ce-pas-1371313

TL;DR as Greg wrote, say pourquoi pas.

In general, it’s hard to predict whether non or pas will appear in a given expression. A good tool to resolve these cases is Google Ngrams.

For example, here’s a comparison of pourquoi non and pourquoi pas showing that pas wins:

But here’s a comparison of non seulement and pas seulement showing that non wins:

So it depends on the phrase.

To do a deeper check if you’re still not sure which one to say after that, I suggest linguee.fr as dimitris mentioned. There you can find translated chunks in English and French and make sure that it’s used the way you think it is. Here’s an example from a search for pourquoi pas :

Et pourquoi pas un vélo ou des chaussures de marche ?
→
Why not a bike or walking shoes?

As Arkeen wrote, when asking why not do a specific thing, you should use the regular negation ne … pas followed by the infinitive :

Pourquoi ne pas poser ta question en ligne ?

Note that while, as already answered, pourquoi pas ? is the natural translation of “why not?“, pourquoi non ? used to be valid and was still be used in the XX^th century literary French.

Moreover pourquoi non ? can still be used in modern French but only in reply of a non and would match “why no?” instead of “why not?“.

– Tu viens ?
– Non !
– Pourquoi non ?

Here, non is just repeated like any other word would be, e.g.:

– Tu choisis quelle couleur ?
– Rouge.
– Pourquoi rouge ? (Why red ?)