The word people aren't sure about 'dago'. Acc. to the OED, it is "supposed to be a corruption of Diego (Spanish equivalent of James)."
The OED: "A name originally given in the southwestern section of the United States to a man of Spanish parentage; now extended to include Spaniards, Portuguese and Italians in general.
1838- 'American, 18 July- "The shrimps... are caught by Dagos."
1890- N. Y. Nation- "Mr. Reed makes no effort to conceal his contempt for this proposition to trade with a lot of "dagoes" as he calls them."
1723- Bumstead in New England Hist.... "The negro dago hanged for fiering Mr. Powell's house."
1832- Wines, in "Two Years in Navy"- "These Dagoes (of Minorca) as they are pleasantly called by our people were always a great pest"
(English and American sailors esp. hated them). Many references could be put here.
1858- Knickerbocker- "And so, Bill, you served as ingineer with these 'ere blamed dagos, you say."
1833- "Old Sailors Yarns"- "We shan't be called away till ten or eleven o'clock when all the dagos are asleep."

The word seems to have been used disparagingly from the beginning. It should be understood, however, that in the 18th and 19th centuries (and beyond), outside of the upper classes (and even there), ANY foreigner was looked upon with contempt by most people from the British Isles (and by extension English-speaking North Americans, Australians, etc.), and given a variety of disparaging names. For the Mediterranean peoples, these names were many.

I doubt that the name originated in the southwestern States; the early reference to Minorcans, etc., suggest a European origin. The 1723 reference also makes the statement in the OED doubtful; at the time the southwestern area of what is now the United States was still under Spanish rule and very, very few gringos were allowed in to trade.
Early reference to 'dagos' in New Orleans (including Sicilians, Corsicans, Maltese and Greeks- thus foreigners in general) suggests that if the term did originate in America, it more likely did so in coastal states with many foreigners.
In 1877, Bartlett, a great historian of the Americas, said: "Dagos. Originally people of Spanish parentage, born in Louisiana, now applied there to Italians, Sicilians, Spanish and Portuguese."

Part of the above from Lighter, "Historical Dictionary of American Slang," vol. 1; the rest from the OED.
(Unfortunately, it seems that vol. 3 of Lighter's magnum opus will never be published since the first two did not make a profit for Random House- a bad choice for a publisher).

The "Gals of Chile" was sanitized. Hugill says he altered both the verses and refrains to make them printable.