...these types of expressions are used so often in Zen Practice.
A key aspect of our way is, for example, to have goals and work diligently, yet simultaneously (on another mental channel, perhaps), to drop all goals and thought of anything to achieve. Such might then be called sometimes "goalless goals" or "non-goals". That is not a goal ... and it is not not having a goal ... so it is a "non-goal". :-)
Perhaps the most basic description is of Zazen, in which we are neither "thinking" (lost in streams of tangled thoughts) nor "not thinking" (a blank) ... so we describe this as "non-thinking" or "thinking not thinking".
Thus, as we drop all needs ... and thus have a need to sit needless Zazen without goal or need ... I called it a "non-need for Zazen".
Much of this is based on the Buddhist vision of the relationship of the "absolute" (all mental categories and divisions dropped away) and the "relative" (this world in which the mind perceives countless categories and divisions, this and that which we label, separate and judge, like and dislike). Our Buddhist practice is a kind of dance of the two in which we come to see how they relate, and how the "relative" is not so divided as we may think. That is the "non" view. Shikantaza, so many of the Koans and such are about mastering that dance ... not just intellectually or as a formula ... but really coming to see and experience that. Even the experience of Phil being a "separate self" is kind of a dream in Buddhist understanding and so there is no "self" ... but, of course, there is our experience of "self" too ... so we might call that, for example, "non-self".