In Game of Thrones season 7 episode 6, When Jon and his group went to the other side of the wall to capture a Wight.
Why didn't they use horses and decided to walk instead? Is there any reason for this?
Horses need a great deal of care & feeding and this was intended to be a quick, quiet, snatch and grab mission.
Equally, the terrain is not really suitable for horses being primarily ice covered rock, mountains and the like.
Out of Universe
Of course, the imagery of brave men walking across a bleak landscape emphasises the desperate nature of the mission and the vulnerability of the whole thing.
Also so that our heroes, once engaged with the Army of the Dead can't just ride away.
The arrival of Benjen Stark on his horse is something of a deus ex machina to allow Jon to survive and return and, of course, therefore breaks all of the in-universe ideas I mentioned above.
What makes you think that horses were available? Eastwatch is manned by wildlings, not by the Night's Watch, and the wildlings don't seem to have or use horses very much. So the simple answer is that they had to walk because there weren't any horses at Eastwatch.
I haven't seen the movie, so I don't know the expected length of the mission. However, infantry will outpace cavalry in marches longer than some five days, as the cavalry mounts need to graze (or they need to carry their own fodder). Unfortunately, fodder is volume intensive and hard grains (which can be used only for part of the horses' diet) are mass intensive.
Also, on long marches the cavalrymen will walk dismounted quite a bit of the way.
As for horses in the cold, I remember only one polar expedition used horses (small poneys), the other expeditions used dogs to pull sleds. Dogs were much better adapted to cold and wind.
The horses are not accustomed to cold and would likely die - especially overnight without stables and food. (That is, if we ignore the ridiculous assumption that the whole trip was an hour or two long so that Gendry could just run back - while the arrowhead-shaped mountain was not visible from the get-go). You don't see the Inuit people using horses. Nor do the wildlings seem use them IIRC.