I can only address Ross Barnett, but in his case there was actually a very carefully orchestrated drama worked out ahead of time between him and his state troopers and the members of the marshals escorting James Meredith onto the Ole Miss campus.

There were a number of secret phone calls between the Kennedys (John and Robert - the president and the attorney general) and Barnett in the days leading up to the integration. Some are archived here at the Miller Center, but they're presented in a more contextualized way at American Radioworks.

The level of choreography was really pretty amazing:

RB: Yes. Hold just a minute, will you? Hello, General, I was under the impression that they were all going to pull their guns. This could be very embarrassing. We got a big crowd here and if one pulls his gun and we all turn it would be very embarrassing. Isn’t it possible to have them all pull their guns?

RFK: I hate to have them all draw their guns, as I think it could create harsh feelings. Isn't it sufficient if I have one man draw his gun and the others keep their hands on their holsters?

RB: They must all draw their guns. Then they should point their guns at us and then we could step aside. This could be very embarrassing down here for us. It is necessary.

RFK: If they all pull their guns--is that all?

RB: I will have them put their sticks down before that happens. There will be no shooting.

RFK: There will be no problem?

RB: Everyone pull your guns and point them and we will stand aside and you will go right through.

You can kind of see a couple parts of the problem there - first, both sides have armed, loyal, uniformed people. And second, these actions were taking place in front of large crowds.

Barnett was an elected leader - he was popular, and had spent months making defiant speeches about segregation and states' rights, in some cases before enormous crowds.

In fact, here - that transcript was recorded on Sept 27, 1962. Two days later, here's Barnett speaking during halftime at an Ole Miss football game - 40,000 people in attendance.

As it was, things got so heated that upon Meredith's entrance to campus a riot broke out that some called "the Second Civil War."

(NPR had a pretty good commemoration of the riot on its 50th anniversary. It's also got its own Wikipedia entry.)

Were the 3,000 U.S. marshals and national guardsmen in Oxford to try arresting Barnett during or after that, I can't really imagine what the public would do, but I can imagine things getting very much uglier.