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I am very sorry to hear about your wife's condition and how these crises were handled. From a legal standpoint, though, it is very unlikely that the facts you have described would typically give rise to any meaningful legal recourse against your employer. The first issue you would encounter is damages. In law, you cannot recover for what "might have" happened, only what actually did, and here, it does not seem that you suffered any actual tangible damages as a result of the way these calls were handled. Certainly the thought of what could have happened is frightening, but that is not a recoverable form of damage.
The second issue is one of proving that your employer owed you a duty and breached that duty. These criterion are at the core of all negligence claims. Generally, employer policies are not interpreted by courts as creating binding legal obligations. They are merely guidelines to give employees an idea of how the employer runs their business and how the employer will act in certain circumstances. In other words, the fact your employer may not have followed protocol for how to handle these phones calls probably does not mean you could sue them for it. Furthermore, actually proving they didn't follow protocol would be very difficult here as you have no way of proving that the first call was impossible to understand.
Now, you have an idea solution to the problem, which is that your wife can now contact you directly on your cell phone. I can't think of anything more your employer can do to enable your access to your wife in times of emergencies.
As far as unemployment benefits are concerned, that is a very difficult question. It is true that an employee can collect benefits if the State determines they quit for "good cause," and it is also true that caring for an ill family member can constitute "good cause" as it is defined in this context. However, these things are judged on a purely case by case basis. Also, since you have an ongoing need to be on-call for your wife, you will need to explain how that need prevented you from working further for your current employer but would not impede you from working somewhere else. This is because proving that you are able to immediately accept new work is a requirement for collecting unemployment. In short, you would really have to thread the needle here, and you would have to take the plunge and quit before you'd have any idea how the state would rule on your claim for benefits.
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.
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