Comparison of meaning: Should vs Need [closed]

I recently saw the following sentence in a recruitment ad:

'9-to-5 employees' need not apply

It's important to say that English is not my mother tongue, and maybe that's why something in the sentence did not sound good to me, because then I was told that the sentence was correct.

I thought that better options would be:

9-to-5 employees should not apply

or

9-to-5 employees need to not apply

Is there a difference in meaning between the three sentences?

Is the original sentence correct?

When need is used as an auxiliary verb, it means that you're under obligation or necessity to do something. Adding not reverses this, it means you're not required to perform the action. M-W says:

be under necessity or obligation to
you need not answer

"need not" is often used somewhat idiomatically to mean that not only is there no requirement, there's no point in performing the specified action. When it says "9-5 employees need not apply", they're saying that they've already decided not to accept applications from these employees, so they'd be wasting their time applying.

"should not" is either a command or strong suggestion, depending on the context. So if they say

9-5 employees should not apply

it means that the applications might not be totally forbidden, they would prefer that they not apply.

"need to not" is a stronger prohibition. It means that not performing the action is a requirement. It's very similar to "must not".