Responses
I believe they were making fun of the mole that's right under his eye. The way the scenes played out seemed to imply that hiding the mole was the reason he had grown out his bangs and was so nervous about cutting them. Since moles on the face are considered "beaty marks" it can loosely be connected to the bullies' insults (although I honestly believe it's one of the dumbest reasons to call someone gay that I've ever come across).
Okay. Something under your eye referring to the crusty stuff that you find under your eyes when you wake up. That is about hygiene and keeping up appearances. Not in any way does it refer to sexuality or whatnot.
Long story short, those bullies need a lesson on morals as well as proper use of words.