Here are the views of W.L. Wilmshurst (one of my favourite authors) who states, there is more in the “square pavement” than simply something for the high priest to walk on.
His paper is as follows:
It is not merely the Jewish High Priest of centuries ago that is here referred to, but the individual member of the craft.
The reason being that every Mason is intended to be the High Priest of his own personal Temple and to make it a place where he and the Deity may meet
By the mere fact of being in this dualistic world, every living being, whether a Mason or not, walks upon the square pavement of mingled good and evil in every action of his life, so that the floor cloth is the symbol of an elementary philosophical truth common to us all.
But for us, the words “walk upon” imply much more than that.
They mean that he, who aspires to be master of his fate, and captain of his soul, must walk upon these opposites in the sense of transcending and dominating them, of trampling upon his lower sense nature and keeping it beneath his feet in subjection and control.
He must become able to rise above the motley of good and evil, to be superior and indifferent to the ups and downs of fortune, the attractions and fears governing ordinary men and swaying their thoughts and actions this way and that.
His object is the development of his innate spiritual potencies, and it is impossible that these should develop so long as he is over-ruled by his material tendencies and the fluctuating emotions of pleasure and pain that they give birth to.
It is by rising superior to these and attaining serenity and mental equilibrium under and circumstances in which, for the moment, he may be placed.
That Mason truly “walks on” the chequered groundwork of existence and the conflicting tendencies of his more material nature.
On occasion brethren have asked me why I make a point of not walking on the chequered pavement in my own lodge room. My answer stems around my feeling that to me it symbolically represents “Holy Ground” and I feel uncomfortable walking upon it.
Having said that, I do not feel this discomfort when, in attending to the ritual of the lodge, and the conferring of Degrees, I am compelled to do so.
I have no idea how to advise Brethren who meet in a Temple where the checkered pattern covers the entire floor.