Yeah, that is confusing. It's an odd typo. It is the predominant employment status in our country and it does apply in your state.
MN law is also "at will." In fact, only one state isn't. Regardless, the state that you work in is the state law that applies. A company headquarters doesn't change the employment law that applies to employees. It only changes the corporate law concerning the business itself.
There is no law requiring a write up process. Employees can just be let go. Now, many employers have chosen to adopt their own progressive termination process, but that is not dictated by external laws. That is an employer created construct. If they have that progressive policy in writing in an employee handbook, it can take on quasi-contractual power, but very few attorneys are willing to take what would be an "implied contract" claim.
So, if the employer has employees with lots of write ups that they are not getting rid of, but are trying to get rid of this person without any write-ups, you can certainly allege some sort of implied contractual basis for keeping here there. This would actually follow with the HR person's comments, because the person obviously believed that there was at least something, somewhere that would keep your friend from being terminated.
But all of that is speculation until you find the company policy and have it reviewed by an employment law attorney. Until then, the default position here is that she is "at will" and can be terminated without cause.