What does that say, other than: that, in a modern sociological environment, certain genetics-mediated brain differences cause people to behave (react) in different ways to their environment?

It says nothing about what inherently is a mental disease and what is not. Perhaps genetic differences causing certain mental "diseases" in average American society (or whatever society in which you conduct a study) actually benefit the individuals in societies with different cultures or structures. In the same way that certain non-psych heritable "diseases" confer some advantages and disadvantages (sickle-cell anemia and malaria, for instance). Fact is, many genetic differences that some people regard as undesirable are not even classified as diseases, even though they have advantages and disadvantages, if not for lifespan, certainly in social mobility within social environments.

tl;dr: it's patently obvious and beneath discussion that in certain genetic differences might lead two otherwise identical individuals to differ on whether they have different behavior, just as they might differ in many other ways. It's the determination that different behavior in some cases is a "disease" that seems to be at the heart of Szasz's critique of psychology.

How do twin studies rule out "environmental" that is, non-genetic, causes. Twins look alike, so if there are ideas about how people who look alike ought to act alike we would be expect a correlation between genes and behaviour that has nothing to do with the content of the genes. Rather, that behaviour would be motivated by similar ideas.

So thoughts, feelings, and behaviors are outside of biology? How does that work?

I realize that you are simply quoting a passage and not necessarily personally espousing what you wrote.

Kendell point (1). Szasz's position is not that diseases must be diagnosed by looking for chemical or structural abnormalities. Rather, such abnormalities are a criterion for having the disease. It may not be possible to find those abnormalities until after the person dies.

Point (2): the existence of such abnormalities is required for a person to have a disease, but not all abnormalities are considered diseases. This is as irrelevant as saying that not all ice cream contains water because some water is not ice cream.

Point (3) is also irrelevant. Szasz's claim is that mental illness is behaviour that a person undertakes for a reason, albeit a reason that people don't like. That idea that such behaviour will turn out to be caused by a disease is like looking for a hardware fault in your iPhone because you have installed an app your dislike.

I addressed (4) earlier. For my reply to point (5) see my reply to point (3).

As for Shorter, note that his statement that mental illnesses are brain diseases is incompatible with Kendell's points since Kendell is attacking the idea that lesions are necessary for disease while Short alleges that such lesions exist. So which position does the poster actually hold? I have already addressed points (1) and (3) by explaining that mental illness is not biological, so let's skip to point (2). Szasz's point was that if a person was found to have a brain disease, brain doctors would treat him with his consent. Psychiatrists don't do this and so there is some difference between psychiatrists and doctors who deal with brain diseases.

Clarke. Point (1): if a person claims to be mentally ill then he is claiming to have an illness when he doesn't. It is better to face up to this squarely and discuss why the person in question is acting the way he acts. If a person claims he is not a moral agent and refuses to face up to his moral disagreements with others, you're not doing him any favours by agreeing.

Point (2): people get labelled as mentally ill because they have a moral problem, that is, behaviour that is regarded as undesirable by somebody who may or may not be the patient. It is true that such problems would not vanish if we stopped calling them mental illness but Szasz didn't claim that they would. Rather, he claimed that we should face up to those problems squarely and stop using quasi-medical excuses to coerce the kind of people commonly labelled mentally ill. If you want to coerce somebody you should have to make an explicit argument that his behaviour should be treated as a crime.

As for point (3), abandoning treatments that the public dislikes is entirely compatible with picking some other set of treatments that the patient dislikes and imposing it on him.