Naruto might feel a sense of responsibility. He would see it as his duty to help Boruto understand and control the Jougan. So, he could start to create special training regimens for Boruto. These regimens would be designed not only to strengthen Boruto's physical and ninja skills but also to help him master the unique powers that come with the Jougan. He would also teach Boruto about the importance of using such power for good and protecting the Hidden Leaf.
In fanfiction, if Naruto discovers Boruto's Jougan, it could change the power dynamics in the Hidden Leaf. Naruto would likely be both proud and worried. Proud because his son has such a unique and potentially powerful ability, but worried because it might attract unwanted attention. He could start to share his own experiences with dealing with unique powers more openly with Boruto. This might include stories about how he tamed the Nine - Tails' power. He would also probably consult Kakashi, his former teacher, to get some insights on how to guide Boruto in harnessing the Jougan's power safely and effectively.
Boruto might be in awe. He has always known his father as the Hokage, a powerful and respected figure. Seeing Naruto's past full of struggles, loneliness, and determination would make him realize how far his father has come. It could also make Boruto more empathetic towards Naruto and perhaps inspire him to be more like his father in some ways.
Seeing Naruto's power in fanfiction can greatly influence Boruto's character. It could make him more determined to reach his father's level. He might start training harder and be more serious about his ninja duties.
It could also be that Naruto sees in Boruto's eye a sign of an emerging danger. The eye might hold a clue to a new enemy or a brewing conflict. Naruto, being the experienced ninja he is, would then start to take precautions and train Boruto more intensively to face whatever this sign in his eye indicates.
One common scenario is Boruto getting to see Naruto in a super intense battle, showing off his amazing jutsu. Another could be Naruto using his power to protect the village, and Boruto watches in awe.
Probably his time at the Ninja Academy. Naruto was always getting into trouble there, but it was also where he made some of his first friends. Also, his encounters with the Third Hokage could be shown. The Third Hokage was like a grandfather figure to Naruto, and their interactions were important in Naruto's development.
Yes, there are. You can find Naruto has Jougan fanfictions on various fanfiction platforms like FanFiction.net and Archive of Our Own. Just search for 'Naruto has Jougan' on their search bars.