The (relevant part of the) 5th amendment does not say "you never have to cooperate with the courts" it says you can not be compelled to incriminate yourself. People and companies are compelled to cooperate with courts all the time - that's literally what a subpoena is.
Btoh you and the justice system. You could plead the fifth on personal incrimination, but that may result in an investigation into your personal matters now that you've indirectly hinted that the facts you were asked to present could indicate wrongdoing.
The system could also consider your testimony to be so important as to proclaim that you shall not be prosecutable for anything that you say as part of your testimony.
Of course, all of this applies only to things within your mind. You cannot use the 5th to get out of presenting any and all existing materials that may even be perceived to be relevant to the court order. That's likely to be as initially broad as emails, business plans and documents, road maps, etc. Obviously, impeding these efforts would be contempt of court or (in the case that you destroy anything) possibly worse.
In practice though, your attorney works with the court and other parties' attorneys to identify what exactly would be relevant to the case and if any special secrecy regarding those materials and testimony is warranted. Case gets worked out, life moves on.
