Texans always repeat this claim. It is not close to being correct!
HAWAII:
Hawaii has a far longer history as an independent nation than Texas. Here is some of the basic overview of the history of the Kingdom of Hawaii:
During the 1780s and 1790s the chiefs were constantly fighting for power. After a series of battles that ended in 1795 and forced cession of the island of Kauai in 1810, all of the inhabited islands were subjugated under a single ruler who would become known as King Kamehameha the Great. He established the House of Kamehameha, a dynasty that ruled over the kingdom until 1872.
The death of the bachelor King Kamehameha V—who did not name an heir—resulted in the popular election of Lunalilo over Kalakaua (Kalākaua). After Lunalilo’s death, in a hotly contested and allegedly fraudulent election by the legislature in 1874 between Kalakaua and Emma (which led to riots and the landing of U.S. and British troops to keep the peace), governance was passed on to the House of Kalākaua.
In 1887, under the influence of Walter M. Gibson, a group of kingdom subjects, members of the Hawaiian government, American and European businessmen forced Kalākaua to sign the 1887 Constitution of the Kingdom of Hawaii which stripped the king of administrative authority, eliminated voting rights for Asians and set minimum income and property requirements for American, European and native Hawaiian voters, essentially limiting the electorate to wealthy elite Americans, Europeans and native Hawaiians. King Kalakaua reigned until his death in 1891. His sister, Liliuokalani (Liliʻuokalani), succeeded him to the throne and ruled until her overthrow in 1893.
Ship's landing force on duty at the Arlington Hotel, Honolulu, at the time of the overthrow of the Hawaiian monarchy, January 1893. Lieutenant Lucien Young, USN, commanded the detachment, and is presumably the officer at right.[20]
In 1893, Queen Liliuokalani announced plans to establish a new constitution that would have replaced the 1887 Constitution of the Kingdom of Hawaii that was established during the reign of King Kalakaua. On January 14, 1893, a group of business leaders and citizens formed a Committee of Safety in opposition to the Queen. United States Government Minister John L. Stevens, responding to a request from the Committee of Safety expressing concern about possible violence directed against American citizens, summoned a company of uniformed U.S. Marines to come ashore. As one historian noted, the presence of these troops effectively made it impossible for the monarchy to enforce its new constitution.[21]
In January 1893, Queen Liliuokalani was replaced by a Provisional Government composed of members of the Committee of Safety. There was much controversy in the following years as the queen tried to re-establish her throne. The administration of President Grover Cleveland commissioned the Blount Report, which concluded that the removal of Liliʻuokalani was illegal. The U.S. Government first demanded that Queen Liliʻuokalani be reinstated, but the Provisional Government refused. Congress responded to Cleveland's referral with another investigation, and submitted the Morgan Report by the U.S. Senate on February 26, 1894, which found all parties (including Minister Stevens) with the exception of the queen "not guilty" from any responsibility for the overthrow.[22] The accuracy and impartiality of both the Blount and Morgan reports has been questioned by partisans on both sides of the historical debate over the events of 1893.
VERMONT:
Flag of the Vermont Republic
Originally inhabited by Native American tribes (Abenaki and Iroquois), the territory that is now Vermont was claimed by France but became a British possession after France's defeat in the French and Indian War. For many years, the surrounding colonies disputed control of the area, especially New Hampshire and New York. Settlers who held land titles granted by these colonies were opposed by the Green Mountain Boys militia, which eventually prevailed in creating an independent state, the Vermont Republic, founded during the Revolutionary War and lasting for 14 years. In 1791, Vermont joined the United States as the fourteenth state.
Vermont continued to govern itself as a sovereign entity based in the eastern town of Windsor for fourteen years. The independent state of Vermont issued its own coinage from 1785-1788[17] and operated a statewide postal service. Thomas Chittenden was the Governor in 1778-1789 and in 1790-1791. The state exchanged ambassadors with France, the Netherlands, and the American government then at Philadelphia. In 1791, Vermont joined the Federal union as the fourteenth state–the first state to enter the Union after the original thirteen colonies.
CALIFORNIA:
California has a very short history as an independent nation, but it is still a chapter in that State's history too:
In 1846, settlers rebelled against Mexican rule during the Bear Flag Revolt. Afterwards, rebels raised the Bear Flag (featuring a bear, a star, a red stripe, and the words "California Republic") at Sonoma.
“ [we] overthrow a Government which has seized upon the property of the Missions for its individual aggrandizement; which has ruined and shamefully oppressed the laboring people of California ”
—William Ide, Declaration from the Bear Flag Revolt
The Republic's first and only president was William B. Ide,[13] who played a pivotal role during the Bear Flag Revolt. His term lasted twenty-five days and concluded when California was occupied by U.S. forces during the Mexican-American War.
The California Republic was short lived. The same year marked the outbreak of the Mexican-American War (1846–1848). When Commodore John D. Sloat of the United States Navy sailed into Monterey Bay and began the military occupation of California by the United States. Northern California capitulated in less than a month to the U.S. forces. After a series of defensive battles in Southern California, including The Siege of Los Angeles, the Battle of Dominguez Rancho, the Battle of San Pascual, the Battle of Rio San Gabriel, and the Battle of La Mesa, the Treaty of Cahuenga was signed by the Californios on January 13, 1847, securing American control in California. Following the Treaty of Guadalupe Hidalgo that ended the war, the region was divided between Mexico and the United States; the western territory of Alta California, was to become the U.S. state of California, and Arizona, Nevada, Colorado and Utah became U.S. Territories, while the lower region of California, Baja California, remained in the possession of Mexico.
In 1848, the non-native population of California has been estimated to be no more than 15,000. But after gold was discovered, the population burgeoned with U.S. citizens, Europeans, and other immigrants during the great California Gold Rush. On September 9, 1850, as part of the Compromise of 1850, California was admitted to the United States as a free state (one in which slavery was prohibited).
OTHER STATES WITH A POSSIBLE CLAIM TO INDEPENDENT NATIONHOOD:
Technically all 13 original States were independent nations in a loose federation at conclusion of the War of Independence. Even if one considers that equivalent of the United States the transition to the Constitution occurred when 9 states signed it. Some background:
The Treaty of Paris, signed on September 3, 1783, ratified by the Congress of the Confederation on January 14, 1784 and by the King of Great Britain on April 9, 1784 (the ratification documents were exchanged in Paris on May 12, 1784), formally ended the American Revolutionary War between the Kingdom of Great Britain and the United States of America, which had rebelled against British rule starting in 1775.
Preface. Declares the treaty to be "in the name of the most holy and undivided Trinity," states the bona fides of the signatories, and declares the intention of both parties to "forget all past misunderstandings and differences" and "secure to both perpetual peace and harmony."
1. Recognizing the 13 colonies to be free, sovereign and independent States, and that his Majesty relinquishes all claims to the Government, propriety, and territorial rights of the same, and every part thereof;[1]
JB continues: The treaty acknowledged for independence of the 13 former colonies. So they were independent nations in federation from that point, 1783 to 1789 when the Constitution went into effect.
History of the Constitution:On September 17, 1787, the Constitution was completed in Philadelphia at the Federal Convention, followed by a speech given by Benjamin Franklin who urged unanimity, although they decided only nine states were needed to ratify the constitution for it to go into effect. The Convention submitted the Constitution to the Congress of the Confederation, where it received approval according to Article 13 of the Articles of Confederation.[9]
Once the Congress of the Confederation received word of New Hampshire's ratification, it set a timetable for the start of operations under the Constitution, and on March 4, 1789, the government under the Constitution began operations.
JB:Virginia and New York joined in June and July of that year, almost simultaneous. However North Carolina stayed out of the USA until Nov of 1789, and Rhode Island until May of 1790. These two states in particular have a claim to totally independent nationhood for this period as the Federation was dissolved and they had not yet joined the new United States of America.