Standard format is from XY onwards, so Break cards are perfectly legal.
As donavannj mentioned they may be referring to cards that "break" the game-such cards that haven't been banned but were classed as such are Seismitoad EX, Trevenant XY, Night March (at one point) and now Battle Compressors and VS Seekers are considered as well.
If it is a rule of the local league to ban Break cards I would consider contacting Pokemon about that as not only does it seem silly but can also hinder practice for Nationals/Regionals/Championships. Or the Professor doesn't know what they're on about.