Typical depiction of a new egg (low entropy state) and a cracked egg (high entropy state), intermixed with discussion on "spontaneity" and the "arrow of time", terms and concepts often used to describe what entropy or the second law means. [3]
In thermodynamics, you can’t unscramble an egg or a "broken egg can't unite back into a whole egg" are oft-used layperson’s descriptions of the either entropy, the second law of thermodynamics, or the arrow of time.

An common logic to this teaching method is that it is said that a fresh new egg is in a "low entropy state" or ordered state and that a cracked or scrambled egg is in a "high entropy state" or disordered state. Then it is said that the second law mandates that systems tend to increase in their entropy, or systems tend towards maximal entropy, or something to this effect.

History
In 1954, English thermodynamicist Alfred Ubbelohde was discussing entropy and irreversibility as embodiment in the examples of Humpty Dumpty, broken eggs, spilt milk, and burnt boats. [5]

In 1955, American physicist Walter Albersheim, in his on “Entropy and Evil”, was using what he called the old saying “one cannot unscramble an egg”. [4]

In 1974, the "drop an egg on the floor", described as being synonymous with "dropping an ice cube into a glass of water" was being discussed. [2]

It is difficult to say, however, who originated either model (dropping or unscrambling), although some references seem to point to the circa 1950s lectures or statements of English physicist Charles Snow, who supposedly commented on unscrambled eggs and entropy, although a source is difficult to locate. It also may have connection to someone who famously used eggs to describe the arrow of time.

Entropy song
The popularity of the 'drop an egg description' of entropy is evidenced by the fact that it is found in the late 1990s parody song entropy by MC Hawking. The specific part of the lyrics explaining entropy is:

You ever drop an egg and on the floor you see it break?
You go and get a mop so you can clean up your mistake.
But did you ever stop to ponder why we know it's true,
if you drop a broken egg you will not get an egg that's new.

That's entropy or E-N-T-R-O to the P to the Y,
the reason why the sun will one day all burn out and die.

The egg model is some time described in term of it “being extremely improbable that all the yoke molecules will wind up finding themselves located in the original location of the yoke.” [1] This statement embodies: the original Rudolf Clausius view that the entropy increase means that particles of the working body will never return to their original state or atomic position (c.1850) plus the Loschmidt’s paradox (1876) + Poincare recurrence theorem (1890).

Difficulty on model
A basic difficulty of this model is that it is largely based on Boltzmann-type gas theory models, where as an egg is not something synthesized via gas particle interactions, but rather chemical reactions. In other words, statistical thermodynamics or statistical mechanics models are used, whereas chemical thermodynamics models are the correct way to explain egg formation. Secondly, the concept that a system has an increase in the value of its entropy, in the original Rudolf Clausius scheme, is a mathematical way of saying that either heat entered the boundary surface or that in one cycle more heat entered the boundary surface than left, in once complete cycle, meaning that some of the heat was converted into internal mechanical work (internal work), according to the principle of the transmission of work and the mechanical equivalent of heat, specifically the relocation of the positions of the particles of the system.

References
1. Ehrlich, Robert and Ehrlich, Gary. (1998). What If You Could Unscramble and Egg? (pg. 156). Rutgers University Press.
2. Sladek, John T. (1974). The New Apocrypha: a Guide to Strange Science and Occult Beliefs (pg. 258). Stein and Day.
3. Temporal problems solve all physics problems (2010) – PhysicsForums.com.
4. Albersheim, Walter J. (1955). “Entropy and Evil”, Rosicrucian Digest, pgs. 467-69. Dec.
5. Ubbelohde, Alfred René. (1954). Man and Energy: Illustrated (pg. 157). Hutchinson's Scientific & Technical Publications.