I got my job through an IT recruiting firm 3 months ago and on my first day I was told that I had to become a 1099. I had never done it before in several years of consulting, but I really wanted the experience this new job was going to offer, so I agreed. The issue is this:

In the 3 months I've been here, I've received 2 checks from a very problematic finance department (1 of 4 I bill each week). The agreement was a net 15 pay structure; however, I've never been paid in the same month, much less within the agreed upon term. As I type this, they are well over a month past due. Emails, voicemails, hand-written letters and escalations through my manager have gone nowhere. I've never even received a response to any of my communications.

So, first, am I eligible to receive unemployment benefits as a 1099?
Second, if I quit given this scenario, will I be covered?

Thanks in advance for any help anyone can give me!

As an IC, you are not eligible for unemployment benefits. Having said that, however, did/do you meet the criteria for independent contractor status? IOW, have you been misclassified (even with your assent)?
http://www.irs.gov/taxtopics/tc762.html

Regarding late payment, you may have to file a civil suit (small claims might be an option, depending upon the amount) if the company has breached the contract regarding payment. It's worth having an attorney review the agreement to determine if, in fact, you have a bona fide contract, whether a breach has occurred, and what your legal recourse may be.