Online accounts of the expedition say that the voyage 'was probably the first official expedition to carry a photographer'. Since photography was invented in Europe not that long beforehand, I'd like to know if the photographs taken during the expedition were indeed the earliest photos ever taken of the world outside Europe. If not, do we know who took the first camera onto a sea voyage outside Europe, and where did they photograph?
While I don't know if any of the previous Royal Navy/Royal Society of London-sponsored expeditions were equipped with cameras, the photos taken by the Challenger expedition of 1872-1876 weren't the first set of photos to feature non-European surroundings, not by a long shot.
The 19th century pictures I'm most familiar with were taken by a Felice Beato, Italian-British photographer and one of the first war photographers. He took pictures of the war in Crimea, and proceed to take photographes of Asian countries such as India(then British), China(then the Qing Empire), Japan, and Korea(then the Kingdom of Joseon). This would have been around 1858-1871, well preceding the Challenger Expedition.
(An account of Beato : https://en.wikipedia.org/wiki/Felice_Beato)
https://commons.wikimedia.org/wiki/File:Image-Secundra_Bagh_after_Indian_Mutiny_higher_res.jpg
1858 India, Felice Beato.
https://commons.wikimedia.org/wiki/File:Felice_Beato_(British,born_Italy)-(Interior_of_the_Angle_of_Taku_North_Fort_Immediately_After_Its_Capture_by_Storm)-_Google_Art_Project.jpg
1860 China, Felice Beato.
https://commons.wikimedia.org/wiki/File:Interior_of_Fort_McKee.,06-1871-NARA-_559259.tif
1871 Korea, Felice Beato.
Also, an Indian photographer named Lala Deen Dayal apparently was commisioned to photograph the governor general's tour of Central India in 1867. Again well before HMS Challenger.
(An account of Deen Dayal : https://en.wikipedia.org/wiki/Lala_Deen_Dayal)
Then, as for the second part. I have to confess that I do not know the answer : when was the first photograph taken outside Europe? If I had to make an educated guess, though, it would be this.
https://www.aramcoworld.com/Articles/November-2015/Capturing-the-Light-of-the-Nile
These pictures of Egypt date to as early as November 7, 1839. Considering the first "invention of photography" was announced at Paris, 7 January 1839, this daguerreotype might be the first picture taken of the world outside Europe. If this is the case, the name of the men you were seeking would be Horace Vernet and Frédéric Goupil-Fesquet, who took photographs of Egypt, for example the Ras el-Tin palace in Alexandria or the Pyramid of Cheops in Giza.