ぬ and ざる

This page briefly explains two older versions of the ない verb ending; ぬ and ざる.*
These endings are more or less the same thing, with very subtle differences that you shouldn’t worry about at this point.

While you are unlikely to see the endings by themselves, there are some phrases that use them (Covered below) that will appear on the JLPT.
ぬ and ざる are NOT interchangeable in their phrases.

*Technically they both come from ず which is an older form of ない。

ぬ

Verb ない → ぬ
する → せぬ
くる → こぬ

This is old Japanese and isn’t likely to be encountered outside of certain phrases.

Examples

• 食べぬ → 食べない –  Don’t eat
• あってはならぬ → あってはならない – That shouldn’t be. (Its bad).
• 何も話さぬ人 → 何も話さない人 – A person who doesn’t talk.
  
  

ねばならない

なければ
Verb ぬ → ねば
する → せねばならない

ねば is an older version of なければ, formed from the ば form of ぬ. (Source)
It is almost exclusively used as ねばならない, an older form of しなければならない.
The meaning is the same, but it is older and not really used today.
However it may still appear in period anime or novels.
This pattern may also appear as ~ねばならぬ or sometimes just ~ねば
ならない may rarely be replaced by some other word, though usually not いけない.

Examples

• もう行かねばならない – I really have to go.
• 食べねばならない時もある – There are times you must eat. 
• 勉強せねば – I have to study.
• 敬語を使わねば殺される – IF you don’t use Keigo, you will be killed.

ざる

Verb ない stem + ざる
する → せざる
くる → こざる

ざる is just another version of ぬ. It tends to be used slightly more often to modify other words.
Meaning wise, it tends to translate as “Un” more than “not”.

Trivia: Due to its similarity to さる, the 3 monkey’s (See no evil…Etc.) names are:
見ざる、聞かざる, and 言わざる  (Unseeing, Unhearing, Unspeaking)

Examples

• 食べざるご飯 – Uneaten food. 
• 帰らざる者たちのため – For the people who won’t come back. 
• 許されざる者 – Unforgiven people (Also the Japanese title for “The Unforgiven”)
  
  

ざるをえない

Have no choice but to do
ざる form + をえない

ざるをえない is the most common usage of ざる you are likely to hear.
It means “unable to not do” formed from ざる (Not do) + えない (Unable)
This must be used about a bad thing.
This is more or less the same meaning as せねばならない

A related expression is やむを得ない which means “Unable to be stopped” or “しかたがない”

Examples

• そうせざるをえない – I have no choice but to do that.
• 皆の前では飲まざるを得なかった – In front of everyone, I couldn’t not drink it.
• 学校側としては退学させざるを得ない (Weblio) – As a school, we have no choice but to force you to be expelled. 

Additional Reading

The information here will not be very helpful, but if you are curious:
ぬ andざる differences:
StackExchange
Chiebukuro (Also referenced in stack exchange post)