An unpaid internship program is subject to some very strict requirements that are difficult to meet. There is a multi-factor test used to determine whether someone can be considered an “intern”. The safest route is to hire students who earn academic credit through the internship program, and to get signed agreements confirming the relationship and parties’ understanding that mirrors the six factors. However, academic credit is not a requirement (nor is it sufficient). The main focus is whether the company derives any benefit from the intern’s activities. This means that the internship should be primarily educational in nature and interns should not do menial work of other employees (make copies, file papers, conduct research, etc.). You can think of the unpaid internship as more of a job-shadowing position with minimal work on general skills (as opposed to company-specific skills) that is closely supervised. I know it sounds very unproductive and strange for such strict requirements, but that is the current state of the law on this issue.
We usually recommend that the company pay interns minimum wage to avoid any issues, but in the end, it is a business decision whether having unpaid interns is worth the risk. That said, many companies probably have less than compliant programs since the risk of an intern suing the company for violation of wage and hour laws is generally low. Some things the company can do to comply with the rules though is to structure the internship like an academic or vocational course, complete with a syllabus or learning schedule. The duration of the internship should be fixed and correlate with the school’s calendar. Interns should be supervised closely – moreso than a regular employee – and not doing work that would have been done by other employees. It also helps to make clear that interns are not subject to the same handbook given to employees (which contains policies like you mentioned, PTO and benefits, that suggests an employment relationship) and if possible they should get a separate handbook with just relevant policies (trade secrets, discrimination, harassment). You can see that what it boils down to is that you want to separate interns from regular employees to the extent possible and to structure the program more like an educational or vocational program than a work internship.
I've attached a form of internship agreement that companies may find helpful