The phrase to walk like an Egyptian means: to walk with arms extended, elbows and wrists bent at right angles, one arm up, one down.

This phrase refers to the system devised by the ancient Egyptians to project the three-dimensional human body onto a flat surface, i.e., with the head facing sideways, the torso facing forward, and the arms and legs facing sideways.

The phrase to walk like an Egyptian was popularised by Walk Like an Egyptian, by the U.S. songwriter Liam Sternberg (born 1949), first recorded in 1986 by the U.S. band the Bangles (i.e., Susanna Hoffs (born 1959), Vicki Peterson (born 1958), Debbi Peterson (born 1961) and Michael Steele (Susan Thomas – born 1955)).

In an interview published in the Akron Beacon Journal (Akron, Ohio, USA) of Sunday 26^th September 2021, Liam Sternberg explained how he created the song:

“I was in a ferry boat crossing the English Channel.”
When the vessel hit choppy water, passengers stepped carefully and moved their arms awkwardly while struggling to maintain their balance.
“It was a pretty tipsy ride and I had too much wine to drink, so I was also pretty tipsy,” Sternberg recalled. “So I just wrote it down: Walk like an Egyptian. It really came from wherever these things do.”

However, the phrase to walk like an Egyptian had already occurred in To Kill a Mockingbird (1962), a U.S. film directed by Robert Mulligan (1925-2008), written by Horton Foote (1916-2009), and starring Gregory Peck (1916-2003) and Mary Badham (born 1952). This is the transcript of the relevant scene in the film:

[Scout sees Jem walking, one arm extended in front of him and one in back of him, and one foot behind the other.]
Scout. What are you doing?
Jem. Walking like an Egyptian. We were studyin’ about ’em in school. Teacher says we wouldn’t be no place without ’em.
Scout. Is that so?
[She begins to imitate his walk.]
Jem. Cradle of civilization. They invented embalming and toilet paper.
[He sees her imitation.]
That’s wrong, Scout!
[He goes to her, kneels and tries to fix her feet.]

The screenplay by Horton Foote is based on To Kill a Mockingbird (Philadelphia & New York: J. B. Lippincott Company, 1960), by the U.S. novelist Harper Lee (1926-2016). However, the phrase to walk like an Egyptian does not occur in the novel. The relevant passage is as follows:

The second grade was grim, but Jem assured me that the older I got the better school would be, that he started off the same way, and it was not until one reached the sixth grade that one learned anything of value. The sixth grade seemed to please him from the beginning: he went through a brief Egyptian Period that baffled me—he tried to walk flat a great deal, sticking one arm in front of him and one in back of him, putting one foot behind the other. He declared Egyptians walked that way; I said if they did I didn’t see how they got anything done, but Jem said they accomplished more than the Americans ever did, they invented toilet paper and perpetual embalming, and asked where would we be today if they hadn’t?

The phrase to walk like an Egyptian occasionally reoccurred before Liam Sternberg’s song was released in 1986. The following, for example, is from Time’s ripe for canned weepter, by the U.S. newspaper columnist John Keasler (1921-1995), published in The Miami News (Miami, Florida, USA) of Wednesday 16^th August 1978:

Canned laughter has become so accepted by most of the American public that the day of canned weepter cannot be far away.
Many daytime television fans are already complaining that they do not know exactly when to start weeping during the soap operas.
I was watching some daytime TV viewers just the other afternoon and it became pitiably evident to me they simply did not know when to weep.
[…]
So the next time the plot took a turn toward a rather somber situation—a blackmailer fleeing police posed as a surgeon and inadvertently sewed a patient’s leg on so backwards he had to walk like an Egyptian—I unobtrusively began to sniffle.
“Sniff sniffle,” I went. “Boooooo.”
I was edging in easy.
(In the soap opera, the heroine who was supposed to marry the guy whose feet faced different ways found out she had parrot fever; both she and her macaw were hot to the touch.)
“Hoo. Boooo hoo,” I went.