I don’t have anything that you have.

And you can change it to I have nothing that you do.

“I don’t have anything that you have.”

This is more idiomatic that saying:

“I have nothing that you have.”

But more important than the above:

DO NOT say “I have nothing that you do.”

This construction is weak with DO as the last word. The “do” is totally ambiguous.

Technically you can get away with the sentence if what you mean is, “I have nothing that you do HAVE.”

But the construction DO HAVE is infrequently used, and the construction DO meaning TO DO is much more commonly used. Therefore to say “I have nothing that you do” implies you’re comparing POSSESSION with DOING.

And besides, no American would EVER say in conversation:

“I have nothing that you do.” This just isn’t done.

Probably best to use a verb like ‘own’ or ‘hold’… “I own nothing that you have” is a lot less clumsy.