Executive orders impact how the executive branch of the government acts.
They are instructions from the head of the executive branch (be it President or a state's governor) for an agency in the executive branch to do a certain thing.
For example: Executive Order 14087 of October 14, 2022 https://www.federalregister.gov/documents/2022/10/19/2022-22...
> HHS Actions. In furtherance of the policy set forth in section 1 of this order, the Secretary shall, consistent with the criteria set out in 42 U.S.C. 1315a(b)(2), consider whether to select for testing by the Innovation Center new health care payment and delivery models that would lower drug costs and promote access to innovative drug therapies for beneficiaries enrolled in the Medicare and Medicaid programs, including models that may lead to lower cost-sharing for commonly used drugs and support value-based payment that promotes high-quality care. The Secretary shall, not later than 90 days after the date of this order, submit a report to the Assistant to the President for Domestic Policy enumerating and describing any models that the Secretary has selected. The report shall also include the Secretary's plan and timeline to test any such models. Following the submission of the report, the Secretary shall take appropriate actions to test any health care payment and delivery models discussed in the report.
This directs the Department of Health and Human Services (under the executive branch) to do certain things.
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https://trumpwhitehouse.archives.gov/presidential-actions/ex...
> Sec. 2. Policy. It is the policy of the executive branch to:
> (a) secure the southern border of the United States through the immediate construction of a physical wall on the southern border, monitored and supported by adequate personnel so as to prevent illegal immigration, drug and human trafficking, and acts of terrorism;
> ...
> (d) Except as otherwise noted, “the Secretary” shall refer to the Secretary of Homeland Security.
> Sec. 4. Physical Security of the Southern Border of the United States. The Secretary shall immediately take the following steps to obtain complete operational control, as determined by the Secretary, of the southern border:
Note again, the Department of Homeland Security is under the executive branch.
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And so, the "could you create an executive order ordering {department} to arrest someone with red hair speaking in public?" The answer is "no" because the ability to arrest someone is under the Judicial branch - not the executive branch.
Secondly, you will note that all of the executive orders are citing the law under which the president (or governor) is given that authority from congress. To arrest a red head for speaking congress would need to have a law (First Amendment!) that allows the executive branch to silence someone.
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I will also point out gag orders which are, well, fairly straight forward censorship. Except that the gag order is from the judicial branch - not the legislative branch.
https://law.yale.edu/mfia/case-disclosed/when-silence-isnt-g...