It is a myth that individuals on the autism spectrum have no feelings. In fact, a number of ASD individuals decribe "hyper-empathy" as if they are a sponge soaking in the emotions of others. Their struggle may have to do with deciphering the specific meaning of the emotional atmosphere, knowing what to do with the emotions of the other person, or staying calm in the midst of another individual's emotions. Many feel the intensity of emotions even when the other person feels they are staying fairly calm. Some ASD individuals describe trying to remain "muted" or withdrawn as a method of protecting themselves from strong emotional atmospheres.
1. Stay calm, steady, and matter-of-fact
Even as I write this, I am in the emergency room with my son. He cut himself and needs stiches. He has already said multiple times that he is more concerned about the reactions of others (me, my husband, and the ED staff) rather than the physical injury or treatment.
I was again reminded that the best thing for him is for me to be very matter-of-fact. The only reason to express my emotions out loud is to release them for my own comfort. He doesn't need to feel more anxiety; what he needs is for me to help him feel calm by remaining calm.
For the autistic individual, whether another person is expressing a "negative" or "positive" emotion doesn't always matter because the emotion itself is overwhelming. Even excitement or concern is overpowering. What many people in the spectrum need is a calm environment, and that can start with those around him/her (and their ability to manage their own emotional states).
2. Plant these seeds for the future
Know that your calm response in every situation will plant seeds for the future that are reassuring to the ASD individual. That is, he/she will remember they can come to you when they feel overwhelmed, and they will not have to also deal with your strong emotions. "It's all I can do to calm my own feelings. I can't reassure my family too."
These techniques would help many situations with those in and out of the spectrum. Remember that your release of emotions impacts those around you.