Henry was later in attendance at the Khitomer Conference, where after the attempted assassination of the Federation president, he was held at phaser-point by Uhura as he tried to escape the conference along with the other conspirators. (Star Trek VI: The Undiscovered Country)
Henry was played by Carlos Cestero and credited in the film as "Munitions Man". The two costumes worn by Cestero, one from the briefing and one from the conference, were both sold off together in the same It's A Wrap! sale and auction on eBay. 
Following the call from Azetbur, where Henry first appears, it is established in the script that "the President and his advisers are a little speechless at her strength and brusqueness."
His character had a more significant role in the film, as evident in the revised fifth draft of the script (dated 15 February 1991), where he was described as a "Munitions Businessman" and called by name by the president. Though all of his dialog was take from an extended scene that was completely cut from the film (including the director's cut), it would have preceded what was otherwise the first of his two brief background appearances in the film.
In the first extended scene, while present in a briefing with the president (following the assassination of Gorkon and the arrests of Kirk and McCoy), he explained to the president that "You have our total support, Mister President. My factories are standing by. We can gear up to full weapons production in less than a week," to which the president replied, "Thank you, Henry. Right now we're pursuing diplomatic channels...," to which Henry acknowledged, "Of course. No one wants a war..." This was immediately followed by a call from the president's secretary announcing the call from the new Klingon chancellor.
Later in the script, while Spock extracted information from Saavik via mind meld (before the character became Valeris), it was revealed that the "Munitions Man" was also in on the Khitomer conspiracy, at which point it flashed-back to him saying "No one wants a war." This too was excluded from the final cut of the film, however, his apparent role remained slightly evident based on Uhura's choice of specifically hold him from escaping from the conference, as was seen as Spock escorted Valeris into the room of attendees announcing the full confession.
According to the novelization of the film, his name was Henry Mulwray.