I suggested that Matthew 16:24 describes a three-step process:
Then Jesus said to his disciples, "Whoever wishes to come after me must
take up his cross, and
(And before any Carmelite gets the heebie jeebies, let me explain that this is a logical process, where the subsequent steps depend on the presence of the prior steps, not a chronological process, where each step occurs in sequence in time.)
I also suggested the first step amounts to deposing one's own will as the center of one's existence.
If these suggestions are sound, then in order to take up your cross it is necessary to depose your own will as the center of your existence. Why might what follows from such denial of self be considered the taking up of a cross?
Well, what does the taking up of a cross signify? I'd say it signifies the somehow voluntary acceptance of a foreseen path of suffering. It's not merely the stoic acceptance of an instance of suffering, but of a whole route of pain and sorrow, a route whose details may be unknown but whose end, Golgotha, can be seen from the start.
If, as the old love song has it, this world is a vale of tears, then pretty much every path through it is one of pain and sorrow. What distinguishes carrying a cross is the willingness (which is not simple resignation) to suffer, and to some extent the choice of a path that does not minimize suffering.
I'd say that what denying yourself contributes to taking up your cross is this: At every moment you bear your cross, you are tempted to set it down. "Setting it down" means bucking against the suffering, or even abandoning the path for one of less suffering. If you have not denied yourself, if your own will is still the center of your existence, two things follow. First, there is nothing but your own willpower keeping you from setting down your cross, and at any moment you might change your mind. Second, the cross you are bearing is in some sense defined by your will; it is a cross you design for yourself, rather than a product of the fallenness of the world itself.
Note, finally, that in what I've written about both denying yourself and taking up your cross, there is nothing explicitly Christian. You can depose your will in favor of all sorts of things -- of the Party, of a charismatic leader, of social forces. And you can choose a path of suffering from all sorts of causes; you can suffer for your Art, or for your children, or for the sheer cussedness of it.
Thus the final step of the process: "Follow Me." If we are following Christ, the first two steps become specific. It is God's will that becomes the center of our existence. It is the suffering of a Christian that we shoulder, with its twin character of internal self-discipline and external buffeting.