No, they cannot have a per se "do not speak Spanish" rule. Federal employment regulations prohibits that. However, your employer is permitted to have a "do not speak Spanish at certain times" rule if there is a justifiable business necessity for it and your employer is allowed to make English fluency a requirement of the job if there is a similar business necessity. I'm not sure what your situation is.
NOTE: This answer is not intended to be legal advice and should not be construed in that way. This answer does not create an attorney-client relationship and no such relationship may be created absent a signed retainer agreement. The author is licensed in Illinois only, and his answer is for educational purposes alone.