I'm not certain what you mean by your first question? Do you mean, did Native Americans know that eclipses were the result of a lunar body orbiting the Earth, which likewise orbits a solar body, interacting periodically for complete or partial occlusion of the lunar and solar bodies visible from certain latitudes from the Earth? I mean, probably not. And given that Copernicus only published his heliocentric model in 1543, we could easily turn this question around to ask whether Europeans at the time of Contact knew what eclipses were and be equally as skeptical of their understanding.
Native Americans certainly wrote about them though! The oldest "book" from the Americas, the Dresden Codex is a Postclassic era work from the Maya Yucatan which predates Contact by a few centuries. In it are famous lunar tables which can be used to predict eclipses through the use of 148 or 177 day intervals. These intervals, which equate to 5 and 6 month lunar cycles, are factors in a larger astronomical cycle of 18.03 years called a "saros," which is the time it takes for the Sun, Moon, and Earth to all cycle back to the same position and thus "repeat" the eclipse of 18.03 years ago.
The 148/177 day pattern can also been seen in the Codex Borgia, which is a Central Mexican text, probably made shortly before Contact, depicting a more Aztec worldview. Moreover, it may actually contain a pictorial representation of a particular total solar eclipse. Susan Milbraith, who in her earlier research had identified Aztec eclipse imagery, writes in her 2013 Heaven and Earth in Ancient Mexico: Astronomy and Seasonal Cycles in the Codex Borgia that the images on page 39 and 40 of the codex not only represent an eclipse, but specifically the 1496 CE total solar eclipse.
See, Mesoamericans were not just tracking lunar cycles and predicting eclipses to pass the time. Such celestial events held deep symbolism as well as astrological observation. The Codex Borgia depiction in fact shows the relationship between solar eclipses and Venus, the latter being notably visible during the former. Borgia shows the Sun, Tonatiuh, being attacked by various deities associated with Venus. Similar associations with the aggression of Venus-associated gods causing solar eclipses are seen in the Dresden Codex which predates Borgia by at least a century.
We should not fall into the ethnocentric trap of assuming Mesoamericans were nothing more than superstitious fools, quailing at the sight of the darkening heavens. While there is certainly a very florid passage in the Florentine Codex about solar eclipses, their symbolism as periods of renewal and liminality between world-states is equally, if not more, important than the literal aspect of the Sun being snuffed out and the world falling into darkness. Aveni and Calnek (1999) suggest there may even have been an element of statecraft associated with eclipses, noting that a number of historically significant (to the Aztecs) events seem to on, or just a few years before or after, eclipses. They also note that Aztec accounts of eclipses do not seem to capture all the eclipses which would have been visible from Tenochtitlan. Putting these two ideas together, Aveni and Calnek put forth the interesting hypothesis that Aztec chroniclers were more likely to mention an eclipse when it occurred near a historically significant event, and perhaps were not above fudging some dates to make certain past events fall closer to an eclipse. Notably, they suggest that the given date for the founding of Tenochtitlan, 1325 CE, was actually retconned by later Aztec writers to fit with a total solar eclipse during that year.
Of course, the main thing you'll probably hear regarding Aztecs and eclipses is that florid quote from the Florentine Codex I mentioned earlier. So here it is in all it's purple prose:
When [the eclipse] came to pass, he [the Sun, Tonatiuh] turned red; he became restless and troubled. He faltered and became very yellow. There there were a tumult and disorder. All were disquieted, unnerved, frightened. There was weeping. The common folk raised a cry, lifting their voices, making a great din, calling out, shrieking. There was shouting everywhere. People of light complexion were slain [as sacrifices]; captives were killed. All offered their blood; they drew straws through the lobes of their ears, which had been pierced. And in all the temples there was the singing of fitting chants; there was an uproar; there were war cries. It was thus said: "If the eclipse of the sun is complete, it will be dark forever! The demons of darkness will come down; they will eat men!"
A bit frightening, right? The passage about superstitions for a lunar eclipse is scary in its own, more down to Earth (heh!), way. Pregnant women would carry pieces of obsidian in their mouths to prevent having deformed children.
Do you mean just those native to what is now the USA, or are you including Mesoamerica?