What’s a “handegg”?

Solution 1:

CBS Sports has this nice article explaining the origin of the word, including a newspaper snippet from 1909:

“Hand-Egg,” Not Football.

To the Editor of The New York Times:

Football is certainly a misnomer, for the game is played not with the feet but with the hands, and the ball is not a ball but an egg.

I propose that the game be played with the feet and with a ball, or else that it be called “hand-egg.”

OBSERVER.
New York, Nov. 7, 1909.

Hand-egg is the proposed word for what has been named football when describing a sport in which an egg is moved using one's hands, instead of an activity where a ball is moved with one’s feet.

Solution 2:

It's a football reference.

• The hat is a football helmet
• The football is egg-shaped and held in your hands
• A touchdown is worth 7 points (including the obligatory point-after-touchdown)

From Wiktionary:

From hand +‎ egg, by contrast to football meaning “soccer”, with the notion that the respective sports are “neither foot nor ball” due to the predominant use of the hands and the elongated form of the ball.

From Urban Dictionary:

American football. As opposed to football/soccer, where players actually kick a ball with their feet, American football involves players carrying an egg-shaped object in their hands.

Solution 3:

A handegg is an American football.

It doesn't (often) touch the foot and isn't (much of) a ball shape.

Solution 4:

Handegg is a village in Switzerland, in the Canton of Bern, somewhat near Lucerne. Beautiful place, I am sure, but just why they named that hat after it escapes me.

Solution 5:

Handegg:

Etymology

• From hand +‎ egg, by deliberate contrast to football meaning “soccer”, emphasizing the use of the hands and an elongated rather than round ball.

handegg (countable and uncountable, plural handeggs)

• (slang, uncountable) A humorous term for the game of American football, or for any other sport called “football” that uses a prolate spheroid instead of a ball and in which the hands may be used, such as Canadian football or rugby.

(from en.wiktionary.org)