Is there any verses in the Al-Qur'an, or in Hadith that's saying to love yourself? Is loving yourself important?
In Islam we believe that psychologically humans are made of two parts. There is the ego (nafs), which is controlled by our brains and genetic instincts. Then there is the soul (ruh) which is where our free will comes from. Adam was a humanoid animal until God breathed His own soul/spirit into him, which gave him his humanity.
[God speaking to the angels:] When I have formed him, and breathed into him of My spirit, fall prostrate before him. (The Quran, verse 38:72)
It is not recommended to love the ego part of ourselves, because this always leads to pride, arrogance and cruelty. Loving this part means that person can be selfish and consider their selfishness something right and beautiful. Without proper spiritual grounding a person’s love for themselves becomes a sort of self-worship where their negative qualities become good qualities.
As for loving one’s own soul, because our souls are in a way extensions of God’s soul, to properly love it one would include God and all other souls in this love. This is the mystical kind of love that so many traditions speak of, and in my opinion it is the only proper way of loving oneself. You direct this love at yourself and then you realize that what you are actually loving is just a small part of a much larger whole, like loving a drop of water and realizing it is part of an ocean.
Love directed at the soul leads to humility and obedience toward God, because it helps us realize our own smallness. While love directed at the ego becomes worship of the animal part of us. Love directed at the soul elevates us because it directs us toward higher ideals (it motivates us to embody those attributes of God that are proper to embody, such as His all-encompassing love and mercy), while love directed at the ego degrades us because it makes us content to be merely our instincts and bodies.
So loving yourself is a very bad teaching when this is mistaken for loving one’s ego. It turns people into narcissists who think they are unique and special and who demand love and attention without doing anything to deserve these. The whole point of loving yourself, if by it we mean loving the soul, is to realize our non-uniqueness, to realize that we are just drops in the vast ocean of God.