I got into it with my boss over a raise during my annual review, then told them the next day that I was picking up a part-time job to make ends meet. I don't blame them for firing me. I was told, though, that "we just aren't in the same place about what we're doing here" but that they're going to have a meeting with me on Monday to discuss me staying on doing my job as a part-time 1099. Is this because they value my work but don't want to be directly associated with me? I wasn't fired for cause, but I couldn't negotiate to keep my salaried position, ending up in this weird middle ground and I don't really understand the logic.
JA:
Whether it's legitimate or not will depend on the actual nature of the work they want you to perform. This page from the NY DOL covers the factors which are considered to make that determination. While the fact that they want you to perform the same work you were previously doing as a normal employee is a potential argument against your being an independent contractor, other factors may outweigh that. It's certainly possible for a company to outsource the function of an employee or a set of employees to a genuine independent contractor, and while it's unusual to do so by hiring the same employee you just fired to do said work, as long as your relationship with your former employer actually changes such that you are truly independent of their control going forward, you might legitimately be able to be classified as an independent contractor.
Whether you want to accept this offer is another matter, of course. If you want to consider it, when negotiating your fee, don't forget to take into account the fact that you will be responsible for more taxes as a contractor, and won't have the same benefits such as health insurance or paid time off that you did as a full-time employee, so your asking rate should be a fair bit higher than your previous effective hourly rate. Don't let them lowball you.
A "part-time" independent contractor position is also unlikely to be considered suitable work, meaning you would likely still be eligible for unemployment benefits on the basis of your termination if you do decide to decline their offer.