Can a private company state in their employee handbook that English is the primary language at that place of employment?

Yes, they can, and they can require employees to conduct business in English.

What they cannot do, is require employees who are on break, on lunch, or otherwise on their own time, to communicate in English. If, for example, two Hispanic employees wish to have a conversation in Spanish during their lunch break, they may do so regardless of any such company policy.

Just needed that verified.