Thank you very much for breaking things down for me.
Whether an employee has resigned is a factual question which turns on the unique circumstances of the situation. You can certainly argue that she "resigned" when she applied for unemployment benefits and indicated to the EDD that she was no longer employed. However, since this notice was not delivered directly from the employee to you and the employee never otherwise communicated to you that she quit, this leaves things unclear. Suffice it to say this is an unusual situation.
Your rights as an at-will employer are expansive. At-will employees can be fired at any time for are non-discirminatory reason not otherwise in violation of CA or federal law. Terminating an employee for poor performance is always going to be lawful unless the reason for the employee's poor performance is a disability which (a) the employer knew about or should have known about; and (b) which the employer could have reasonably accommodated.
For example, an employee with a severe carpel tunnel syndrome cannot be fired for slow typing ability ("poor performance") if the employer could purchase voice reocognition typing software for a reasonable price that allows the employee to type quicker without using their hands.
The circumstances you describe remotely implicate disability discrimination, but it's a reach. Obviously, you can't stop an employee from making unreasonable or unspported legal claims, but a plausible claim for disability discrimination arising from termination of an employee in this circumstance is very unlikely, especially because you have witnesses and documentation of her poor attitude and work ethic. No disability excuses those things.
As far as continuing healthcare goes, unless you employ 20 or more individuals you have no obligation to provide continuing healthcare coverage once an employee quits or is let go. If you have 20 or more workers, you likely have an obligation under the Consolidated Omnibus Budget Reconciliation Act to keep the employee on your group plan and allow them to pay their own premiums at the reduced group plan rate until they find a new job. See here for more information: http://www.dol.gov/ebsa/faqs/faq_compliance_cobra.html
As far as how to proceed moving forward, you have a few options. First, you can formally terminate the employee (make sure to pay their final wages immediately if you do this) stating as the basis for your decision poor performance. Alternatively, you can construe the notice you received from the EDD and this employee's application for UI benefits as a resignation and leave it at that. Finally, you can keep them "on the books" and try to improve relations once they return, though this option does not seem very realistic under the circumstances.
If I were in your circumstances--and I am speaking purely hypothetically here--I would opt for construing their UI application as a resignation. This gives you the strongest position to contest this employee's claim for unemployment benefits and also makes it hard for the employee to make a claim for wrongful termination, since you never technically "let them go."
As always, please let me know if I can be of any further assistance. I will be unavailable between the hours of 6:00pm-11:00pm, but will be online for the next half hour and later tonight.