Henson asked that anyone with information about his sister call the (910) 433-1114 hotline police have set up.
Bordeaux hasn't been seen since an acquaintance said he dropped her off near her apartment early last Saturday morning. She was officially reported missing to police when she didn't show up for work Monday.
Henson said the family still hopes to find her alive.
"Nobody is giving up hope," he said. "We're still looking every day."
Henson told reporters he insists on keeping a positive outlook and didn't want to hear any negative questions.
"There's no doubt in my mind we're going to find her alive," said Henson. "Anyone who wants to ask a negative question - we're going to have a problem."
Henson said the family had just received another thousand missing person fliers and he's been spending 8-12 hours a day walking through the woods looking for his sister.
"It's so unimaginable to wake up every day and this is the reality," he said.
Henson also dismissed any talk that his sister has simply gone AWOL - saying people leave the military without permission, but they don't leave their family.
"You're not going to not contact your family," he said.
Police have focused their search in the area around the Froggy Bottoms bar on Ramsey Street. That's where Bordeaux was last seen in public the night she went missing. They're also looking up the Interstate 295 corridor near the bar because they said that's the last area where Bordeaux's mobile phone pinged a cell phone tower.
At a news conference Thursday, investigators said they'd checked out a tip about a pond off Dobbin Holmes Road, but a dive team discovered no new evidence in the case.
Police say they've confirmed that Bordeaux's husband of two years was out of town when she disappeared.
The Orlando Sentinal reported he was in Florida visiting his father when Bordeaux was reported missing.
Police have twice questioned a registered sex offender about Bordeaux. In exclusive interviews with ABC11, 25-year-old Nicholas Holbert said he was at the bar with Bordeaux and later drove her home.
Holbert helps out at Froggy Bottoms and lives in a lean-to behind the building. He said he first met the woman earlier this year.
"She came into Froggy Bottoms a couple of weeks ago with a friend - male friend - and they was drinking and playing pool. And I asked if I could get next on the pool game. They said yes and that's how I met them, playing pool at the bar," said Holbert.
Holbert said he didn't know Bordeaux was married until police told him.
"I asked her if she was single, and she said 'Well, it's complicated,'" said Holbert.
Asked about the comment at Friday's news conference, Bordeaux's family said they didn't know what it meant, but pointed out they don't know if Bordeaux really said that.
"That's hearsay," said Henson.
Henson declined to talk about Holbert further.
Holbert said the only thing he's guilty of is making a friend. He maintained Bordeaux asked him to bring her to Froggy Bottoms Friday night where he helps tend the bar. He said something unusual happened when he drove her back to her home.
"Around one, one-thirty, she told me 'I'm tired. I want to go home.' I said 'Okay,' so we got in the car and as soon as I pulled into Meadowbrook, she said 'You stop right here and let me out. I'll walk,'" said Holbert.
The spot Holbert said he dropped off Bordeaux is about a quarter mile from her apartment on Water Trail Drive. He told ABC11 he figured there might be someone at the apartment she didn't want him to see, so he left.
Holbert said while Bordeaux was at the bar, she spent a lot of time on her cell phone.
"I don't think she called anybody, but she was texting at her phone all night to somebody," he said.
Holbert said he had nothing to do with the soldier's disappearance and wishes he could talk to her family.
"I'd just tell them, I'd tell them I'm grieving too. She seemed like a really nice and cool person. And I hate that she's missing and hope that she is found for their shake so they can get closure," he offered.
Holbert told ABC11 he believes he's been targeted by police because of his criminal record. He's a registered sex offender who was convicted as a 16-year-old of indecent liberties with a 5-year-old child.
Holbert has not been charged in Bordeaux's disappearance.
He spoke again with ABC11 Thursday after his second round of police questioning. He said one thing investigators asked him to do shocked him.
"He said 'Look in that camera and state if I killed Kelli Bordeaux then I deserve to die,'" said Holbert.
"You did that?" asked ABC11's Gilbert Baez.
"I did that," Holbert said.
"Did you kill her?" asked Baez.
"I did not," said Holbert.
"Did you have anything to do with her disappearance?" asked Baez.
"I had nothing to do it. I picked her up and took her home. That's it," said Holbert.
In a third exclusive interview with ABC11 late Friday, Holbert admitted he failed to properly register his address as a sex offender with authorities and he was turning himself in to deputies on that charge.
He said the arrest was completely unrelated to the Bordeaux case.
Officials said Holbert is being held on $100,000 bond.
Family members described Bordeaux as 5' 1" tall and weighing 102 pounds. She has blond hair and brown eyes. She was reportedly last seen wearing a pink tube top with spaghetti straps and black shorts.
The Army said Bordeaux, whose full name is Kelli Marie Bordeaux, is assigned to the 601st Area Support Medical Company, 261st Multifunctional Medical Battalion, 44th Medical Brigade, as a health service specialist.
Bordeaux, a native of St. Cloud, Fla., joined the military in April 2011. She arrived at the unit in November 2011.