This is mostly a question of context: "Du wirst mich nie dazu bringen, zu reden" is something that would be fitting in a context where the speaker is calm but wants to be absolute clear that he never will be talking.
"Du wirst mich nie zum Reden bringen can be used in the same Situation but isn't quite as powerful, also it is fitting when screaming at someone in anger or while being tortured.
"Du wirst mich nie zum Reden zwingen" as well as the last one is the same as the previous, the change of the word "bringen" to the word "zwingen" at the end would be likely due local customary behaviour.
Much more uncommon but still done, especially in literature and movies:
The change between "zwingen" and "bringen" can change the meaning of the sentence.
The following sentence:"Du wirst mich nicht zum reden zwingen, aber für ein kleines Sümmchen könnte ich dir etwas verraten"
English: "You can't force me to talk, but for a small amount of money, I could tell you some things."
In this case it is not so much about the talking but the ability to force someone to talk, "bringen" can't be used in this way.
But only if you have the context, we be able to tell you how exactly it is meant or to be used.