Everyone has to make choices: agree or disagree, protest or accept, Coke or Pepsi. Eisenhower's executive order was a direct response to those who chose not to follow a federal law. He used the word "wilful" (or "wilfully") three times to emphasize that some folks made a deliberate decision to disobey the law. It also implied that lawbreakers were responsible for the actions that followed their decision.
In a way, Eisenhower no choice but to order the schools to integrate. As president, he'd sworn to uphold the Constitution, and the Constitution said that the schools must be desegregated. He said as much in his initial response to the Brown decision: The Court has spoken and I must obey. The choice, though, was how he'd do this, and he chose to put the full military force of the executive behind the integration effort.
Remember, y'all: choices have consequences. Ike made his, knowing full well that lots of his southern constituents and politicians would come down hard on him. His Little Rock decision didn't hurt his re-election numbers, though—it was an electoral landslide. And when did you last hear of a Republican Presidential candidate getting 40% of the African American vote?
Decisions about schools and other community matters are best left to local majority rule.
Once a law is settled, the locals have to accept it, like it or not.
The heart of the Brown decision and Eisenhower's obligation to enforce it is the Equal Protection Clause of the Fourteenth Amendment: "No State shall […] deny to any person within its jurisdiction the equal protection of the laws."
After emancipation, the southern states issued laws called "Black Codes" that severely restricted the civil liberties of freed Blacks. The justification was that the codes would restore order and protect blacks from the consequences of their own "laziness and ignorance" (source).
The Fourteenth Amendment was ratified in 1868 to grant citizenship to freed Blacks and to protect their civil liberties. The southern states refused to ratify it, so Congress put the South under military rule and passed the Reconstruction Act , which pretty much told those states that if they wanted back into the Union, they'd better ratify the Fourteenth. They did, but soon found ways around it: how about separate schools, trains, bathrooms restaurants, and clubs? As long as they're, you know, equal. That stuff lasted until the Civil Rights Act of 1964.
You should totally fall in love he Equal Protection clause of the Fourteenth Amendment; it's one of the most important guarantees of our system. It's the basis for how American society functions; the law protects everybody. Desegregation, equal employment opportunities, the Americans with Disabilities Act, voting rights, the right to a public defender, same-sex marriage—all those have the Fourteenth Amendment to thank.
If you ever visit the Supreme Court Building in Washington, D.C., you'll see engraved over the entrance: EQUAL JUSTICE UNDER LAW. When you stop and look at it and really think about what that means, guaranteed it will blow your mind.
The Fourteenth Amendment was necessary because some states would never have granted equal rights to African Americans.
Because the Fourteenth Amendment also says "All persons born or naturalized in the United States […] are citizens of the United States," it's become part of the ongoing fight over the rights of immigrants.
It's a glorious thing to be a Pirate King.
But being president isn't too bad, either. Sure, you've got Congress and that annoying checks-and-balances thing to deal with, but if you just want something to happen, you can get it done yourself—within constitutional limits, of course.
How? The Executive Order.
You can give federal employees a day off, establish new government agencies, impose sanctions on nasty countries, free the slaves, throw Japanese Americans into interment camps, or order the desegregation of a high school by the Screaming Eagles.
Opponents of Executive Order 10730 called it an abuse of power that denied states their traditional right to deal with public education. There's a general consensus that presidential power has increased since the founding of the U.S. There are ways that Congress and the Judiciary can reverse an Executive Orders, but that's a pretty rare occurrence. More often, they're just revoked by a new president. Every president issues executive orders, and the opposition party always calls them illegal and unconstitutional, and implies the prez is acting like a dictator.
P.S. Hey, current prez: think you could executive-order every American a pair of these?
Eisenhower believed he had a "clear obligation to act" in Little Rock, even if that meant using armed troops to enforce the law.
Faubus believed the rights of Arkansans who didn't want to integrate were being ignored by the federal government.