1. What does it mean?
헌내기 replaces 새, new, in 새내기 with 헌, old or used. Students use it playfully for themselves after the fresh-newcomer stage, especially when they feel more familiar with campus routines.
Also written or said as heonnaegi
2. The nuance
It is not an official academic year or a claim that someone is old. People draw the boundary differently, and self-directed use is usually lighthearted.
3. Examples
“나도 이제 헌내기라 길 안내는 할 수 있어.”
I'm no longer a freshman, so at least I can give directions now.
“새내기 행사 보니까 헌내기가 된 게 실감 나.”
Seeing the freshman events makes me realize I'm not the new kid anymore.
Sources
References support the meaning and usage. Example sentences are written for NoCapWiki.