Well, I'll kind of go in reverse to answer your questions the best I can. First, your question 'are there actually any examples of the US people using their right to bear arms to influence their government, or even rising against the government?' - the answer is yes. Very early on, just after the U.S. really became a country, you had Shays' Rebellion, as well as the Whiskey Rebellion. These were small-scale "people's revolts" that were at least partially armed by the private citizen-soldiers themselves. You could argue that the young country's provision that its citizens be allowed to own firearms aided in these revolts, yet even these micro-rebellions realized that they needed more firepower than what they yielded on a person-to-person level. Trying to secure weapons from federal armories was almost always the priorities in these engagements, which sort of precludes the notion that private citizens wielding their own weapons had any real influence on affairs (except, perhaps, on a psychological level, in that the weapon ownership empowered these people and made them more prone to armed rebellion).

You had John Brown's attempted slave-revolt in Harper's Ferry (1859), which you might point to as another example of a citizen using their private firearm ownership to influence the government or affect change. Still, this is also an example of how a collection of civilians, fairly well-armed, stand no chance against organized, trained, and almost-always-better-armed troops.

The American Civil War, while an honest-to-goodness rebellion, was not kicked off because private citizens had access to rifles and muskets. It was because state governments very formally organized a government, and unified thousands of scattered groups of soldiers into formal armies under a central command. They also raided federal arsenals in the weeks leading up to (and just after the declaration of) war, and kept up manufacturing of weapons throughout the war (the Tredegar Iron Works in Richmond was especially productive).

I only bring this up to emphasize that while rebellions have indeed taken place in the U.S., they were not the exclusive product of citizens and their Constitutional right to bear arms. This was a negligible, somewhat ancillary byproduct of larger issues at play.

As to your initial question about why the 2nd Amendment (right to bear arms) is such a hot-button topic here in the U.S., that is an immensely complicated question that can spin you down any number of philosophical, sociological, and historical rabbit holes. From the history side of things, I will say that the US's use of the 2nd amendment goes back to the English Bill of Rights (1689), which was influenced by King James II's attempt to keep his subjects unarmed (and less-prone to revolt). Once the English finally got over on James II, they wrote it into their law that citizens could, by right, carry weapons. The Founding Fathers liked this idea, and wrote it into our Bill of Rights. It's hard to know what their intent was in putting it in there, as scholars have argued endlessly if it was meant purely for militia purposes (so citizen-soldiers only), or broadly for all U.S. citizens.

The point is, U.S. citizens have a right (within reason: we can't just own rocket launchers...without some serious paperwork) to own firearms. It is in our Constitution's Bill of Rights. A lot of very sane, reasonable individuals in the U.S. are arguing for limits on gun ownership. They are trying to extend the reach of background checks, limit the sale of firearms at gun shows (which demand fewer checks and roadblocks to purchases), prohibit the sale of large magazines, and even ban the sale of certain firearms altogether (ones that can be easily modified to full-auto, for example). Steps like these would limit the amount of carnage that people can unleash on unsuspecting victims while still allowing responsible individuals to purchase sporting rifles and even handguns for home defense.

Large interests groups funded by the gun lobbies, and most prominently, the National Rifle Association, work hard to block this legislation, and to stir up nasty public sentiment against even the slightest attacks on gun ownership. These people have very deep pockets funding them to make sure people will always have access to guns, and have found that when there's a notion that the guns might be taken away, people buy even more!

This is only the most broad of answers, as any number of people with any number of different political agendas will tell you variations on this story. If you want to read more about the origins of the Civil War, I might go with Bruce Catton's 'Mr. Lincoln's Army' paired with James McPherson's 'Battle Cry of Freedom.' To get some great information on the post-Revolution social landscape of the U.S., I might try Robert Coakley's 'The Role of Federal Military Forces in Domestic Disorders.' I'd defer to a 20th century Americanist on recommendations for the growth of the NRA and gun lobby during the latter half of the 20th century.

[EDIT - Not an hour after posting, I found this article that talks about a recent N.R.A. effort to combat gun-related legislation in Seattle, WA: http://news.yahoo.com/nra-sues-over-seattles-adoption-gun-violence-tax-181145804.html]