Don't show him what are in the box!

The entire embedded question

what [is/are] in the box

is a noun clause serving as direct object for the main clause’s verb show.

This embedded clause has its own subject, which here happens to be what; when it’s the subject of a finite clause as it is here, what by itself is always singular and so demands a singular verb. Correct agreement therefore demands is here, never are.

Try it with other embedded question clauses. Consider questions like:

Who is on the phone?
What is next?
Which is best?

Now imagine that the X here is that same embedded question just given:

I’ll show you X.
I don’t know X.
I wonder X.

That yields things like:

I’ll show you who is on the phone.
I don’t know what is next.
I wonder which is best.

When the qustion word is not itself the subject, you have to look at the actual subject’s grammatical number to pick its verb:

What are your friends doing?
Which ideas are discarded?
What noises are you afraid of?
How is she feeling?
How are you feeling?
Whose parents are missing?
When is she calling?
When are they calling?
Why is she calling?
Why are they calling?

Now embed those, taking care to undo any inversion where needed:

I’ll show you what your friends are doing.
I don’t know which ideas are discarded.
I wonder what noises you are afraid of.
I’m not sure how she is feeling.
I’ll ask how you are feeling.
I can see whose parents are missing.
I always know when she is calling.
I can tell you when they are calling.
Let me show you why she is calling.
I wonder why they are calling.

Sometimes you can get an implied plural subject when the interrogative is acting as the determiner of an implied subject that’s been omitted because it was just used.