Okay, thanks again.
The general rule in the state of Washington is "at will" absent an express agreement to the contrary. At will employment can be terminated for any reason not amounting to discrimination on the basis of a legally protected trait (race, religion, gender, etc.) or retaliation for engaging in certain forms of legally protected conduct (filing a wage claim, taking FMLA leave, etc.). It doesn't matter whether the basis for termination is fair, reasonable or even true.
No law requires you to resign simply because you have been asked to resign, but no law would prevent your employer from terminating you under the circumstances you describe, given the applicability of the "at will" employment doctrine. So, if you refuse to resign and leave it at that, there is a strong possibility you will simply be fired.
As for unemployment benefits, you typically lose your eligibility to collect such benefits if you resign. In order to collect, you would need to prove that you resigned in lieu of IMMEDIATE termination. In other words, resigning did nothing to cause or accelerate your unemployment. You can certainly submit your resignation in writing and state that you are resigning in lieu of immediate termination. This would create a papertrail that would help you get approved for benefits.
Alternatively, you could work with your employer and let them know that you are willing to voluntarily resign (which will disqualify you from unemployment, most likely) in exchange for them giving you a set period of time to look for a new job while you are still employed. This sort of negotiation may be the best approach if your goal is preserve employment as a means of increasing your probability of finding a new job.
I hope that you find this information helpful. Please do not hesitate to let me know if you have any questions or concerns regarding the above and I will be more than happy to assist you further.