Have you seen the title of this piece? It seems innocuous enough, and perhaps even a little boring, but a lot of really fascinating stuff hides behind it — so when my editor asked me if I'd like to write something with this title, I got quite excited. Why? Hyper empathy syndrome has been described in the scientific literature, but not extensively. It's not a diagnosable condition under the current version of the Diagnostic and Statistical Manual of Mental Disorders, and the same holds true for Asperger's — which, though very real, is now officially just autism spectrum disorder. And what about empathy? Aren't autists known for not having much of that, but quite erroneously?
Why Is Asperger Syndrome Officially No Longer A Thing?
The American Psychiatric Association (APA) decided to no longer include Asperger Syndrome in the latest version of the US "psychiatrists' bible", the fifth edition of the Diagnostic and Statistical Manual of Mental Disorders (DSM-5). Instead, the form of autism formerly known as Asperger's became part of a wider diagnosis of "Autism Spectrum Disorder", and is now considered to represent the "upper end" of this new diagnosis. 
The decision was far from widely welcomed, however, and it's highly unlikely that the term "Asperger's" will disappear from popular vocabulary anytime soon, or that those who would formerly have qualified for this diagnosis will stop referring to themselves as "aspies". In addition, the International Classification of Diseases, Tenth Revision (ICD-10) — used for diagnostic purposes in many countries outside the US — still features Asperger Syndrome .
Neurotypical folks tend to define Asperger's by what they believe aspies lack:
- Issues with understanding verbal and non-verbal communication, often taking things too literally.
- Strange behaviors like repetitive mannerism, having a hard time coping with change, and hyperfocusing on particular interests.
- Seeming insensitive to other people.
- Seeking solitude and not wanting comfort from others when distressed.
- Difficulty befriending people and maintaining friendships. 
Aspies themselves see things differently. Here's a sample of symptoms of Asperger's as some aspies may describe them:
- People call you “sad” for being interested in interesting stuff.
- You feel “different” from most people, and feel that you don’t “fit in”.
- People think you’re being rude and/or critical when you’re not meaning to be.
- You hear a lot about how “you’re only making things worse for yourself”.
- You laugh later, and more loudly, than everyone else. [4 — and there's more where that came from!]
I mention this because rather a few people suspect they have Asperger's without ever having been formally diagnosed. These people may or may not benefit from a formal diagnosis (which would, as we've seen, differ depending on where they live), but those who are pretty sure they're aspies may still benefit from stuff written for people with Asperger's even if they've not been diagnosed.
OK — Now For The Empathy Part
Two broad kinds of empathy are generally recognized as existing:
- Cognitive empathy is the ability to accurately "read" another person's emotional state. This is sometimes talked about as the ability to "put yourself into another person's shoes."
- Affective empathy is a deeper state of "feeling with" the other person, experiencing their emotions with them. 
Some people would add compassion to that list, and define it as empathy coupled with undertaking steps to help a person who needs help.
So, where does the idea that people with Asperger Syndrome lack empathy come from? From the scientifically supported fact that aspies often have difficulties with accurately recognizing emotions in others — difficulties with "reading" another person's mental state well. Research shows that people with Asperger's take longer to realize that someone is sad than neurotypical folks, for instance. What about the affective empathy part, then? Studies reveal that aspies don't have any problem with that. It may take aspies a bit longer to realize what a person is feeling, in other words, but people with Asperger Syndrome are not cold-hearted, devoid-of-empathy, people. [6, 7]
Here's an excerpt from one ridiculously fascinating paper titled "The Intense World Syndrome – an Alternative Hypothesis for Autism":
"We [...] propose that the autistic person may perceive [their] surroundings not only as overwhelming[ly] intense due to hyper-reactivity of primary sensory areas, but also as aversive and highly stressful due to a hyper-reactive amygdala, which also makes quick and powerful fear associations with usually neutral stimuli. The autistic person may well try to cope with the intense and aversive world by avoidance. Thus, impaired social interactions and withdrawal may not be the result of a lack of compassion, incapability to put oneself into someone else's position or lack of emotionality, but quite to the contrary a result of an intensely if not painfully aversively perceived environment." 
Does That Mean That People With Asperger's May Have 'Hyper Empathy Syndrome'?
No, insofar as only one case of hyper empathy syndrome has really been documented in the medical literature — the fascinating case of a woman who had part of her amygdala (a part of the brain that processes emotions) removed in an attempt to relieve her epilepsy. The woman could subsequently not just recognize other people's emotions with almost frightening accuracy and "feel with" other people, but also physically experienced the effects of other people's emotions. 
Hyper empathy syndrome might have been diagnosable as "Personality Disorder Not Otherwise Specified" in the previous version of the DSM, and the current DSM-5 just may cover it as "Personality Disorder Trait Specified" , but one of the diagnostic criteria is that the symptoms must not be explainable by other factors. If hyper empathy is an inherent part of your autism, that means you can forget about that particular diagnosis.
Leaving official diagnostic criteria behind, it is, however, absolutely possible to suffer from hyper empathy — something some people refer to as being an "empath" — if you're an aspie. You may intensely experience other people's emotions and be overwhelmed by the "vibes" they give off to the point it causes genuine suffering. You may even experience panic attacks as a result. You may also, on the other hand, experience extreme and positive connectedness with humanity as a whole. Learning about how to control negative and positive emotions when you have hyper empathy syndrome may help you some, though.