As far as I have been able to determine mequano is not an actual drink. The only reference I have been able to find is the one you quoted. It should be noted that if it were in fact a Hawaiian drink, mequano would not be its native spelling. The Hawaiian alphabet does not use the letter "Q". The only letter in the Hawaiian alphabet that could be used for a "Q" like sound is "K". Searching some alternate spellings turns up nothing. So it appears that, to be blunt, Boothby was making stuff up. Having said that though, there is a kernel of truth to his description.

The native Hawaiians do have an indigenous distilled drink. It is called okolehao (literally "Iron bottom" or "Iron rump"). The origins of okolehao are not entirely clear. As it is known today it likely dates to the 19^th century, though it is not really possible to know with certainty. The standard story I have found regarding its origins is that the Hawaiians had an alcoholic drink utilizing the sugars produced from the roots of the ti plant for fermentation. When whalers began to frequent the islands, the Hawaiians acquired try-pots and, combining them with gun barrels, made rudimentary stills, producing the first batch of okolehao

As I said, the production of okolehao started with the roots of the ti plant. They would bake the roots in an imu which caused the roots to produce sugar. The baked roots would then be added to water to let ferment. After leaving the concoction to ferment for a few days they would then distill it.

As time went on the making of okolehao gained some variation. The main variant I've found to be attested is using fruit, hops, and water. They would mix the fruit (in the sources I've looked through they list pineapple and oranges) with the hops and water and let that ferment. This mix appears to have been called "swipe", at least in some communities. Once the "swipe" had fermented they would distill it. Another drink (sorry, I'm not sure what its name is. Source doesn't say.) utilized koji rice, sugar, and hops. They would let that ferment in wooden barrels. This drink was not distilled though, rather it was drunk basically straight from the barrel (after being filtered).

It is worth noting that in the interview with John De Guair Sr., he mentions molasses as being a potential ingredient of okolehao. He says that you could make it out of "anything that ferments." I did not come across the mention of using coffee in okolehao, though.

That's what I was able to find on Hawaiian alcoholic drinks. Boothby potentially based his recipe loosely on okolehao but mequano itself appears to be, at least as a Hawaiian beverage, apocryphal (and I still have no idea where he found that name).

Sources

A note on my sources. Most of these are interviews with various Hawaiians from the 1970s and 80s. All the interviewees were born in the early 1900s. The various transcripts are hosted though the University of Hawaii. They don't really address the history of the drink but rather contain an oral account of how the interviewees participated in or saw the drink being made. All the interviews only devote a couple questions to okolehao though, it is not a main focus. Pretty fascinating reading regardless.

Burton, Erik. "Ancient Kauai Mapping Project: Using GIS to locate and map ancient Hawaiian agricultural landscapes on Kauai." Ethnobotany Research and Applications 9 (2011): 349-377.

Gutmanis, June, and Julia JB Bryant. "Oral History Interview with Julia Bryant (JB)." (1978).

Gutmanis, June, and Tito Marciel. "Note from a Recorded Interview with Tito Marciel." (1978).

ʻHodges, Iwalani, and Benjamin Kapena Kalama. "Interview with Benjamin Kapena Kalama." DU629. W44 W3 1988(1986).

Komagata, Faye, and John De Guair Sr. "Interview with John, Sr. De Guair." DU629. K65 S63 (1981).

Lee, Vivien, Yukie Yoshinaga, and Joseph Batalona. "Oral History Interview with Joseph Batalona." (1978).

While waiting for an answer here, you may have better luck at /r/AskFoodHistorians which is more targeted to that niche.

Did you try /Cocktails ?