When used as an euphemism for foutre, "ficher" has an irregular infinitive (fiche instead of ficher) and an irregular past participle (fichu instead of fiché).
This is generally explained by the influence of foutre (past particile foutu). This is satisfactory for the participle, but I've always found the form of the infinitive a bit baffling (why not fichre or fichtre? - that last one exists but only as an old fashioned interjection, like "golly" or "by Jove", as mentioned in Jylo's answer). It looks like the 19th century innovators of fiche interpreted foutre as a stem /futR/ + a suffix /ə/, and backformed fiche (/fiʃ/ + /ə/) from it. I have to accept it as the only satisfactory explanation I've encountered, but I must confess it's a process that seems very unintuitive from my 21st century viewpoint and with my final /ə/ dropping dialect.
Your teacher was thus indeed correct when they offered "Qu'est-ce que j'en ai à fiche ?" as a bowdlerized version of "Qu'est-ce que j'en ai à foutre ?" What's more, the regularized version "*Qu'est-ce que j'en ai à ficher ?" that your propose is just plain incorrect and might confuse a native speaker hearing it.
Likewise, "J'ai rien foutu de la journée" (I didn't do shit all day) can only be replaced by "J'ai rien fichu de la journée" and not by "J'ai rien fiché de la journée" (A correct sentence, but with a different meaning)