Thomas Szasz seems to be popular in certain Popperian circles, so I shouldn’t be surprised to find that they corresponded and Szasz saw his own views as rooted in Popper’s philosophy. Their mail exchanges can be found here.
Interestingly, Popper, who generally loved Szasz’ books, felt he couldn’t publicly support Szasz, in part because he had two objections, one practical and one theoretical:
The practical point is Free trade in drugs. At the risk of hurting you: This is just silly. (Principles cannot be taken to their limit.) Do you wish free trade in hydrogen bombs and cobalt bombs? I am definitely against free trade in shotguns, even in pistols. Even if I could be persuaded (I might be) by your argument that it is “none of the governments business what drug a man puts into his body”, it is precisely the main business of government what he puts into another person’s body (a dangerous drug, a bullet, an electric shock). That is the practical point.
The theoretical point is the Non-existence of mental disease. Here my difficulty is that I know very little about the subject and you know a lot. …I think that you are 95% right! A third point is the danger of this problem degenerating into a quarrel about words (or “definitions”). But I believe that if a man is heavily drunk, he does lose, as commonsense puts it, “control of himself”. No doubt there are many other drugs that have similar effects, including drugs produced by our own body when it is ill. …I am in my 83rd year, and my memory is getting bad (a mental illness, in my opinion — but let us not quarrel about words). My mind shows symptoms likely due to aging — the aging of the body, presumably analogous to the pains in my joints. … This is the kind of thing that I and most people would call “a mental illness on a physical basis”; and I am not prepared to quarrel about words or definitions. I am very ready to believe you when you say that most alleged “mental illnesses” are not real, that most or all neurotics are not really ill (but undisciplined). But my (fortunately not yet catastrophic) loss of memory is not merely a loss of mental discipline; and it seems to me that it has a physical basis (as, perhaps Parkinson’s disease seems to have).