You need to let go of this idea that energy is like a substance that can be contained in things, or that things can be made of energy. Energy is just a number, an accounting system. There is no energy inherent in anything. Energy is just a number (or a set of numbers with different labels) that we can calculate for a system.Yes, I understand the explanation of zero energy in the box, but what of the energy inherent in the particle?

Nothing is needed.What is it that pushes the particle around, its own energy?

Newton's first law of motion: a particle that is not acted on by any force will travel at constant speed in a straight line.

Nothing is needed to keep something moving in a straight line at constant speed. Things just do that, all by themselves, without needing anything to push them around.

No energy is needed, let alone an infinite energy. A constant input of energy is not needed to keep an object moving at constant speed.So what infinite energy keeps the particle in the box from ever coming to rest?

No. For it to have infinite energy it would need to be inside the walls of the box. The infinitely strong force prevents that from ever happening, in the simple model of the particle in the box. Between the walls, the potential energy is zero, so particle has a finite amount of energy, equal to its kinetic energy, which doesn't change.I am trying to connect "Infinite potential" with "Infinite energy". Does the particle has infinite energy?

Well, maybe, but it's impossible to get any useful work out of an "empty" cubic centimeter of space.* So, even if that's true, in practical terms it doesn't seem to matter very much.Bohm actually proposed that a cubic cm of space contains as much energy as a trillion atom bombs.

The Energy of a Trillion Atomic Bombs in Every Cubic Centimeter of Space!

Michael Talbot and David Bohm (in quotes) in Talbot'sThe Holographic Universe, Chapter 2: The Cosmos as Hologram, p.51

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(* The Casimir effect might complicate this statement a bit.)