How do you say "and all the things that break you are all the things that make you strong" in french?

"et toutes les choses qui vous détruisent sont toutes les choses qui vous rendent fort"

et toutes les choses qui vous cassent sont les choses qui vous rendent fort.

You're right in your translation of "vous cassiez." That's not what you're trying to say. You want "vous" to be the object of the verb "casser," not the subject of that verb. The subject of the sentence is the plural "choses" which means you need the third person plural of the verb "casser," which is "cassent." Also, "forte" is the feminine form of "fort," and there is no special reason for you to use the feminine form here. Masculine is "fort."

et toutes les choses que vous cassiez sont toutes les choses qui vous rendent fort

Is correct, I speak fluent French google translate is normally wrong.

Tout ce qui ne tue pas rend plus fort.

It's a Nietsche quotation, so you can't make a word for word translation.