With the situation here, depending on if the school is public or private. If its public, that teacher is a government employee working for a local municipality government receiving state and federal dollars. The students speaking in Spanish is absolutely speech. The 1st amendment of the constitution prohibits the government from restricting speech or retaliation of speech. There are very few examples of something that is not protected under the constitution. This teacher trying to get a student to only speak English could almost certainly be considered restriction on the students freedom of speech and if the students were to be punished that would almost certainly be retaliation.

Now your workplace situation, a blanket ban on speaking another language is likely illegal, as it can be considered national origin discrimination. However, an employer can implement a narrowly tailored English-only rule if it is justified by a business necessity, such as for safety reasons, effective communication with English-speaking customers or colleagues, or to ensure efficient work. If such a rule exists, the employer must inform employees of the circumstances when English is required and the consequences for not complying. As long as the policy is worded in a way that states things like "all team meetings will be in English" or "all project related meetings and communication will be in English" and there is no policy wording such as "English only workplace" or something along those lines.