Robert Richards, 33, returned from a monthlong stint in Mississippi the day after the shutdown began. He said he was owed about $2,500 in expenses. “We’re tired of being put in the middle,” he said.

Though Mr. Trump said on Twitter over the weekend that “most of the workers not getting paid are Democrats,” that is far from true in places like Jackson County, Fla., where Marianna is the county seat. It is a Republican bastion so deeply conservative that it was illegal to sell liquor by the drink until November 2017. The president and his plan for a wall along the border are popular here, as they are across much of the state, which might explain why Florida Republicans in Congress have done little to pressure party leaders in the Senate to put an end to the shutdown.

“Everybody I talk to wants the wall,” James Grover, 72, a car salesman from nearby Blountstown, said over breakfast on Saturday at the Waffle Iron, a diner on Route 90 that opens six days a week even though its facade, destroyed by the hurricane, is temporarily made up of plastic sheeting and plywood.

Few prison guards interviewed leveled any criticism at the president or his border policy, instead blaming the impasse on both Republicans and Democrats in Congress who have failed to reach any agreement.

“You can point fingers at both sides,” said Jason Griffin, 44. “I point fingers at everyone. If they want to get something done, they can.”

[Here’s a look at what is and what isn’t affected by the government shutdown.]

Mr. Vinzant, the union president, said he believed a wall was necessary because he trusted fellow public employees who work for the Border Patrol. “Those guys will sit there and say, ‘We need help,’” he said. “So I have to agree with it. We don’t have a choice.”