Many years ago when I was still in High School, one of my History teachers told us that the Civil War was bad for us in terms of international relations. Other countries saw us going to war as hypocritical due to the way we separated from England. And, in fact, as I recall, he quoted the following passage from the Declaration of Independence: "That whenever any Form of Government becomes destructive of these ends, it is the Right of the People to alter or to abolish it."
According to my teacher, the Federal government's policy was destructive to the South's economy and way of life, thus they had the right to leave the Union, as per one of our own country's founding documents.
Is there any truth to this at all? Was preserving the Union seen as hypocritical by anyone, foreign or domestic? Googling this hasn't turned much up.
Firstly, the Declaration of Independence is a founding document, but it's not a law. It is occasionally cited in rulings by courts, but it doesn't grant any rights in itself. Anyway, the answer to your question is yes, certainly there were people who thought that the Confederacy had as much right to become independent in the nineteenth century as the US did in the eighteenth. Confederates themselves compared their cause to the American Revolution all the time. They also compared themselves to Italy, which had recently unified, become independent, and been recognized by European powers and the United States; or to the 1848 revolution in Hungary, which was crushed by Austria. Alexander Stephens, the Confederate vice-president, argued that Lincoln was hypocritical for having supported the Hungarians but waged war on the Confederacy; Democrats in the North made the same argument.
It was much easier for Union supporters to argue that the Civil War was a war for the liberation of slaves after the passage of the Emancipation Proclamation, so that really took the wind out of the sails of this argument, particularly in Britain, which was vehemently anti-slavery. It's worth reading the debate in Parliament over the recognition of the Confederacy to see what British politicians were saying about the Civil War. The resolution failed. The general conclusion both by modern historians and contemporary observers is that Bright (pro-Union) destroyed Roebuck (pro-Confederacy).
I want to add that it seems like your high school teacher was making the rather common argument that the Civil War was Not Really About Slavery, that it was actually about industrial policy and tariffs and so forth, and that victory of the North over the South represents the victory of industrial capitalism over a premodern, agrarian, knightly society. This argument is not in particularly good favor among academic historians, although it used to be. I think it's fair to say that these days, most scholars of the Civil War would say that it was ultimately about slavery.
There are a ton of books about every single aspect of the Civil War, including its international ramifications. Some recent ones I'd recommend are The Revolution of 1863 by Andre Fleche, A World On Fire by Amanda Foreman, and The Cause of All Nations by Don Doyle. I used the first two as sources for my comment here. I haven't actually read all of the last one, which just came out this year, but it looks good. A good book on the Confederacy is A Shattered Nation by Anne Sarah Rubin. Lastly, I really recommend listening to David Blight's course on the Civil War and Reconstruction. Seriously, if you want to have hours of super interesting history fun, listen to them, please.
ETA: Oh yeah. There are other reasons that the Civil War caused problems with foreign countries that weren't directly related to whether they found the Union hypocritical or not. For example, the Trent affair, which involved a Union officer who snatched two Confederate diplomats off a British ship. The British were not pleased, for obvious reasons.