Two different kinds of material are often mixed together in a deletion demand like this. The first is work product and business information, such as marketing assets, customer data, or internal files. Employers generally have a stronger claim to those, and a policy against sending them to a personal account can be a legitimate reason to ask for deletion. The second is material about the employee personally, such as their own performance reviews and their own communications with HR about a manager's conduct. Whether an employee may keep copies of those documents, and whether sending them to a personal account was permitted, depends on the employer's policies, the employee's agreements, and the law of the state, so there is no single answer. The wording of the demand still tells you a lot. A message that asks for deletion of 'all' emails and for screenshots as proof does not distinguish between these categories, and it arrives with compliance staff copied, which suggests the company is treating the matter as a formal compliance event. If a legal claim is realistic, deleting relevant material before getting advice can create problems that are hard to undo, and many lawyers will want to review the situation, and may send a preservation request to the company, before the former employee responds. This page explains how to read the language. It does not tell you which documents you may keep.
Rules on employee access to personnel records vary by state, and some states give current or former employees a right to request copies. Rules on whether forwarding company information to a personal account violates the law or only company policy also vary, and may involve confidentiality agreements, trade secret law, or computer access statutes depending on the facts. Courts have also looked at whether an employee who retained documents in connection with a discrimination or retaliation complaint acted reasonably, and results differ by jurisdiction and facts. Because these questions turn on location and the specific documents, confirm the rules that apply to you with a licensed employment attorney in your state.