Q. What is the reason for the custom to say that someone has “yener machla” instead of saying he has “cancer”?

A. Horav Shlomo Miller’s Shlit’a opinion is that this reflects a Talmudical principle (Brochos 19a). The Talmud relates that, during the levaya, the mourner would stand and justify Hashem’s judgment, saying: Master of the Universe, I have sinned greatly against You, and You have not collected even one one-thousandth of my debt… then Abaye asserted that a person should not express himself in that way, as Rabbi Shimon ben Lakish said, and it was also taught in the name of Rabbi Yosei: One must never open his mouth for the Satan to find a way to accuse him, i.e., one must not leave room for or raise the possibility of disaster or evil, since Satan may bring it on him.
See also Shulchan Aruch (Y.D. 376: 2) that a visiting dignitary should not tell a mourner or the ill person he is visiting, and who want to stand up in his honor, to stay or remain as they were, since that could be interpreted, that he wishes they remain in their sickness or their mourning.
The Rov added that someone justifying his absence at an event he was expected to attend, claiming being unable because of health issues, does not constitute an act of “Al iftach peh,” since people are often not feeling completely well.

Rabbi A. Bartfeld as revised by Horav Shlomo Miller Shlit”a