"らない" or "ません" ending particle are often used to indicate that what has been expressed in the sentence is the speaker's will.
e.g
食べません。
I don't not eat
I don't want to eat.
I will not eat.
寝ません。
I will not sleep
行きません。
I don't want to go.
I will not go.
I don't go.

Exceptions:
知りません。
I don't know
思えません。
I can't believe that's the case
食べられません。
Inedible
足りません。
not enough

So, "フランス語は全然喋らない。" is closer to "I will not speak French" than "I cannot speak French" or "I don't speak French".

There is a twist, however, some people use "喋らない。" or "ない" or "ません" ending particles in senses much closer to "I would not" or "I cannot" or "I don't" or sometimes it's used just to make the sentence more indirect and ambiguous.

This is just my personal speculation,so please don't take this at face value. Anyway, I sometimes feel that as some people are using formal speech less, they can't or would not use Keigo, some ending particles have gained new meanings and are being used as substitutes to particles or expressions considered formal.

To be honest, personally I would consider ”フランス語は全然喋らない。”to be an odd sentence because, to me, it means "I will not speak french" and that is an unusual thing to say as a language is not a thing one do or don't or will or will not but can or cannot speak(At least in Japanese).

Times may have changed, however, some people seem to use "フランス語は全然喋らない。" to mean "I don't speak french". And "フランス語は全然喋らない。" is a pretty literal translation of "I don't speak french" too.