I have been unemployed for one year and collecting unemployment benefits. I have been offered a temporary position working under a 1099. My question is if I accept this position and it terminates will I be eligible for unemployment benefits either based on my previous employment or as a 1099 employee.

No. You would not be eligible for unemployment as you were an independent contractor not having any taxes taken out of your pay and not paying into the UE fund which would enable you to receive the UE benefits.

Agreed; however, a job being paid under Form 1099 is STILL a job. In this economy, beggars cannot be choosers.