In 1793 Captain George Vancouver, as a gesture of goodwill, presented King Kamehameha I with a Union Jack. The British gave the Hawaiians other gifts as well, and as a result there was a great deal of respect between the Hawaiians and the British. In fact, Kaumualiʻi, king of Kauaʻi and Niʻihau before becoming a vassal of Kamehameha in 1810, wished to be called King George, and named his son George after the British monarch. As a result, the Union Jack served as the unofficial Hawaiian flag between 1793 and 1816.
During the War of 1812, Kamehameha wished to remain neutral, but it was difficult to do so when he had the same flag as one of the combatants. So, in 1816 a new flag was designed (some say he designed it personally, while others say he merely commissioned it). This is the flag that was flown by the first Hawaiian ship to sail to a foreign nation, China.
Just in case anyone chooses to quibble over whether it's the "Union Jack" or the "Union Flag", there is strong evidence to show that the flag was referred to as the Union Jack long before the jackstaff even existed:
http://www.flaginstitute.org/wp/british-flags/the-union-jack-or-the-union-flag/
The problem is there were many stories running around as Hawaiians had no writing system and only recorded their history by verbal hula chants to tell their past. Many Hawaiian separatists today claim the original Kanaka-Maoli Flag is said to have been Kamehameha’s I personal flag long before the modern Hawaiian flag. British navy Captain Lord George Paulet destroyed it when he took control of Hawaii for five months in 1843.
The Royal Standard of King Kalakaua was displayed in 1881 looked like this
The Hawaiian Crown flag of King Kamehameha III 1833, said to be first flag of the Monarchy which hung at the Kings home in Nuuanu, Oahu.
You have to understand the King Kamehameha I was adept at plying politics as well as being a warrior king. He often flew the Union Jack as to be polite and show loyalty when the British came. Especially since they earlier had provided privateers warships to help his conquest of Maui and Oahu both by transport to keeps his warriors fresh, and to provide cannon and muskets for fighting. Also the British protected the Islands from the French who were at one time trying to force their wine upon the kingdom.
Other times King Kam I would fly the British flag, and when they left and the Americans came. They flew the American flag instead.
When Hawaii became a territory the flag became this
Today the state adopts the Union Jack and the bars of red, white , and blue to represent ties to both England and USA as a courtesy due to historic portions intertwined in its history.
FYI Captain Cook Monument in Kealakekua Bay on Big Island Hawaii. Its one quarter acre parcel is the only owned non-embassy British land in the USA. A British warship stops by once a year to clean the monument.
Strange but true fact. The soil directly below the statue of James Cook at Kealakekua Bay on Hawai'i Island is British sovereign land.
http://www.nauticapedia.ca/Articles/Cook_Memorial.php
"The land under the monument was deeded to the United Kingdom in 1877 and is considered as sovereign non-embassy land owned by the British Embassy in Washington DC and managed and maintained by the British Consul–General in Los Angeles California."