I'm not familiar with the specifics of their argument (frankly I don't really want to know) but the Constitution conspicuously lacks an exit clause.
Edit: If I had to argue for legal secession, I would say that people have a natural right to self-government, and therefore to secession if they feel that their interests are not represented adequately in the national government. A natural right cannot be abrogated even willingly and any agreement that purports to do so is invalid, in the same way that a person's right to self-determination prevents him from rightfully being made a slave in any circumstances, even if he were to willingly sign a contract to that effect. But that's probably not what they're going for...
Ultimately the strong rule over the weak but even then it was considered in bad taste to do so without layering some abstract moral principles on top first.
I actually do believe in moral principles despite being facetious, and I think opposition to slavery is a very powerful principle. I'm a lot less impressed by preserving the union. Frankly I don't understand how it motivated ordinary northerners to fight. If they weren't abolitionists then why did they care whether or not the southern states seceded?