The origin of the word "Gringo" has at least a couple of theories.
One is that it is a corruption of the word "griego" (Greek), because around the time of colonization, the Spanish were fighting the Turks for control of the Mediterranean, and the Christian allies they did not understand were the Greeks. Over the centuries, the word morphed into "gringo".
In Argentina, it refers to anyone not a Spanish-speaking Argentine.
Another, less plausible theory is that as the American settlers came into Texas (they were originally invited, if they would swear they were Catholics and agreed not to engage in slavery) they sang the old English song "Green Grow the Rushes-O".
"I'll sing you one, oh, Green Grow the Rushes-oh.
What is your one-oh.
One is one, and all alone, and ever more shall be it so."
It is a question and response song that goes up to twelve-oh.
Neither the words Gringo or Anglo bother me.
The word you do not want to be called is "Yanqui"
The Yanquis are the ones they want to go home.