What is the ruling on reading the Qur’an in public places, such as at work, and is it permissible to interrupt the reading to return the greeting of a non-Muslim?

It is permissible to recite the Quran in pure places. It is also permissible to interrupt the recitation to return a greeting or to answer someone who asks a question, because continuing the recitation is not obligatory. It is legislated to return the greeting of a disbeliever by saying "wa 'alaykum" (and upon you too) if they offer a genuine greeting.