The "Shiny Clause" is a fan modification to the catching rule of the original ruleset.
"If you encounter a Shiny Pokemon at any point in your run, (first encounter or not,) you may catch it.
While the Shiny Clause is a modification of the catch rule, many people will say "catch but cannot use" but in the clause statement you are allowed to release the first encounter ONLY if it is the same Pokemon. This way, the extra catch does not actually affect the challenge. (This does not apply if the Shiny is in fact the first Pokémon you encounter in the given area. Meaning you CAN use a Shiny, even if it is the second encounter.)
The simple rarity of Shiny Pokemon makes them so desirable, many players are inclined to take advantage of such an uncommon event, especially since it may never happen again.
Most Nuzlockers accept the Shiny Clause as a legitimate modification of the original ruleset, (especially if the extra catch is forbidden from use,) so runs that utilize the Shiny Clause are still considered true Nuzlocke Challenges (they are allowed into the FCR section of the official Nuzlocke forums).