Fear of losing can make you overly cautious and unwilling to take any risks that would help you checkmate. If you realize you have reached an overly cautious state of mind it's again time to take a break and mentally prepare next time to be more courageous. If feeling unsure it's time to play freestyle or other forms of chess as a warm up. Just to loosen up.
Both over confidence and being over cautious kills winning chances. Chess is a mental exercise where the most balanced players (defense/attack), mentally prepared, skilled players with good speed can win (or loose despite everything). I think it's good to be proud even when loosing if you didn't make mistakes and played well, then it was not your fault and the other was just better. I do think over confidence is worse because it can make you blame others instead of improving yourself. Overly cautious can be more patient and study more - better long lasting effects on tactics. But if no confidence attacks are difficult to start. Balance is key.
Sometimes I feel players lose not because of lack of skill, but because of emotions. Fear makes you play too safe, while greed makes you over-push. Which one do you think destroys more games?