Mostly there's no truth to it. At the very least it's not that clear cut and your acquaintance is playing fast and loose with some important complexities, whether they're aware of the fact or not.

The legal situation is unclear, but as a practical matter the tendency of historians is toward the argument that they did not have a right to secede and most of their contemporaries agreed with that. Though neither secession nor a perpetual union are spelled out as features of the Constitution, only reading the former into it was controversial at the time.

That's not the only way to parse rights, though. What I've just written applies to a formal, legal right to secede spelled out in a law book or document somewhere. Most nineteenth century Americans who look at the question do not believe that states have that kind of right. Those who think otherwise are often honest enough to admit that they're inferring it from historical circumstances or constitutional silences rather than the clear letter of the law, with John C. Calhoun being the prime example.

What is much more common is talk of a revolutionary right to secession. A legal right is one you can go to court to defend if someone interferes with it and hope for a judge to agree. It all operates within a legal system and in a fairly orderly fashion. Even Calhoun's formulation of secession doesn't really foresee that kind of outcome. (His is all about brinksmanship and daring the feds to blink first, when you get right down to it.) A revolutionary right is a different animal entirely and nineteenth century Americans were not shy about pointing out the main example of one they had: the Revolution. The colonies ultimately did not have a right to levy war against their lawful sovereign, but they asserted the right to revolution all the same. The test of the right of revolution is not the outcome of a court decision, but rather the verdict of war. If you win your war, then you post facto gain a rhetorical right to have fought it. If you lose, you didn't and might expect to all hang together. With the legal situation being at best unclear -Calhoun's argument is dodgy and wildly impractical despite being about the best that the South ever had to offer- and the revolution failed, there's not a lot left.

All that said, the early nineteenth century has a whole lot of talk about disunion. You'd think from an unexamined read of that discourse that some kind of consensus existed that a state could do such a thing as an ultimate last resort. Closer reading shows that a lot more (and less) is really going on there. Elizabeth Varon identifies five senses of the concept that are useful for understanding the texts.

Disunion as prophecy is the simplest. It's predominantly the cry of self-conscious moderates who want to silence the politics of slavery because they fear it's going to lead to the destruction of the nation. It's raised specifically as a calamity to avoid, not by people advocating it. (They come later.) As such, references well into the 1840s usually come with an immediate disclaimer that this would be terrible and they hope that God and good sense save the nation from such a disaster.

Disunion as threat is a close relative, but not quite the same thing. The context here is that the other side (usually Southerners complaining about antislavery, but the other way does happen too) has been so aggressive, despotic, and domineering that it's a dishonor to the other party and they need to consider some kind of redress. It serves a similar role to threatening an affair of honor, where the real goal is to establish your resolution, convey how serious this all is, and get the other side to back down. Along the way, one also shows one's manhood and so wins plaudits back home. This isn't just empty posturing, which is important for our question. The threat carries with it the declaration that a war over the issue is just and worth fighting.

In the 1830s and 1840s, things begin to heat up more. Immediatist abolitionists (end slavery right now with no compensation to the enslavers) hit the national stage as a new movement and the rhetoric shifts to Disunion as accusation. Now the argument becomes that the abolitionists are actually plotting disunion and wrecking a previously tranquil (for white men) nation with their fanaticism. Either they are going to drive the South to disunion that way or the mutual recriminations are eventually going to bring about widespread social discord that will probably be nation-ending. In the 1840s, some abolitionists flipped the script and argued that proslavery men were the real disunionists, not content with the slave power conspiracy that already rules the nation and ever demanding more and more.

Disunion as process is a bit hazier. One of the best statements of it is in Calhoun's final address to the Senate, where he marks out how continuing and increasing differences among white men over slavery inaugurated a kind of death spiral of mutual distrust and recrimination. Disunion, from the mid-1830s, can look like something that's happening in the public mind in a way it did not before. Even if no one were consciously plotting it -we'll get to them- increasing sectionalism requires new means to contain it, like Calhoun's constitutional theories or else the nation is doomed. But the other side can rightly point to those ideas as a new kind of dangerous radicalism which is itself contributory to a disunionist spirit of increasing prominence in white discourse.

In the 1850s, we see the fruits of all of this in the South in the form of Disunion as program. Until this point, most disunionists, most of the time don't view it as something anyone is going to actually do at least in the near term and possibly not ever. Even when Calhoun lays out a system for it in the 1820s and 1830s, it's a conservative system designed to preserve the Union with secession only as an absolute last resort. Southerners come increasingly to argue that Yankee disunionism, which they decide is unconstitutional, has all but wrecked the Union already and what they need to do is get out of it on their own terms and before it's too late. If they can manage that, they can control the course of Disunion and transform it from a chaotic process into one that proceeds in an orderly fashion and preserves the slavery central to their civilization.

There are secessionists before this point, but they are few and rarely heeded. It's only the final antebellum decade, when Southern demands and successes provoke a genuine huge response in the North, that actually pulling the trigger becomes something conceivable for most white Southerners. They know they're taking a massive risk. They know this can go badly wrong. But ultimately a critical mass of them cannot deal with the emergence of a North that, while still vastly less united on antislavery than the South is proslavery, is united enough to no longer accept fig leaves over general capitulation in every slavery fight. At that point, constitutional theories once wildly heterodox become a sectional orthodoxy and what they've "always" believed in. Disunion becomes a patriotic duty, a way to get away from the chaotic and destructive process that the North wants to subject them to.

It did not go to plan.

You may be interested in this answer by /u/Borimi