If you are fired for refusing a flu shot that your employer recently now requires, can you be denied unemployment?

Yes, you could and should be disqualified from receiving unemployed comp, since you willfully failed to obey a reasonable request. Employers may require this.

It is unlikely, but may depend on knowing more facts. In a healthcare or caregiver role with immune compromised or frail patients, or daycare job with kids, for example, it may be known or obvious from the start that you would need to have a flu shot. Similarly if they told you at the beginning of the job, it's more likely that benefits could be denied. It's more likely that you can get benefits if you can explain a good reason why you don't want the shot. This is a situation where you should really consult with an expert, either by calling a lawyer or the Unemployment Law Project. It's often hard to answer questions like this without more information about the situation.

The State unemployment office will make that decision.

Probably, unless you can prove you did it for religious or health reasons, i.e. you are allergic to flu shots.