Yes, your teacher is right. It should be "hadn't done."
Song lyrics are not the best medium to learn the English language as they are prone to commit grammatical errors, sometimes to match the literacy level and language ability of the character/protagonist. For example, if the theme character of the song or movie is not shown as highly educated, the script will correspond to his/her literacy level. Also, song lyrics contain slang/informal expressions not followed and approved by professional class of English language users. Therefore, it's not a good idea to rely on songs/music to hone one's English skills.
And guess what? There is one more error in that song quote!
"There's many things.." should be "There are many things.."
Why? Because the singular verb form "is" does not seem to agree with the plural subject "many things." So the final correct sentence should be:
There are many things I wish I hadn't done.
Bonus: Check this article for songs with bad grammar!
Bad grammar songs
Enjoy!