Any one can send someone to collections for just about any reason if money is owed. Often times the collection agency will require proof of the debt if you ask for it which is probably why HR is going to send you the evidence now. It would be the same evidence a collections company would ask for. There is nothing the collection company can do if you choose not to pay the debt except to keep hounding you and putting it on your credit report. Different uses of the credit report can matter (car insurance checks it in the calculation of your rates, car loans and home mortgage interest rates can depend on your credit, some potential employers may check it depending on the job position, etc) That is the big stick that they hold -- how it can affect your future credit.
To get a garnishment on wages though, the employer would have to go to court and get a judgment against you. And it is possible in that case that they wouldn't win the case due to reason,timing, etc. The court may decide it is a mistake the employer has to eat. (that's assuming CA allows for garnishments for reasons such as this -- I work in a state that protects employees from most garnishments -- just child support, back taxes and student loans)
Will they do either? you can never know until it happens. It could just be a threat.
But under my personal belief system, if they could prove their mistake and I indeed owed them and they were willing to work out some type of payoff schedule, I would do my best to do so. I know it is hard when you are in school and on student loans but I would hope since it was their mistake that they would be willing to work with you.