Preventing secession isn't mentioned or implied anywhere in the Constitution as an inherent right. You can think that secession should be illegal for these reasons, but there is no legal framework backing it up.
Indeed, it's not mentioned. It doesn't need to be. A Bank of the United States isn't mentioned either, but it's still Constitutioinal. Hamiltonianism at its finest.

The Federal government has a duty to enforce Federal laws on the states as well as to secure national defense and integrity. It can therefore do anything necessary and proper to prevnt the dissolution, and destruction, of the united states. Allowing wanton secession contradicts the prime directive of a national defense. Secessionists are literally enemies of the State and domestic criminal revolutionaries. It entails forceful separation from the rest of a country by entities that gave up their full sovereignty long ago.
Secession could merely be another inefficiency in the federal government put in by the Founders like separation of powers and federalism.
Unfortunately, the Father of the Constitution has just the opposite interpretation. Have you read Madison's view on Secession? He clearly states there is no intended, or implied, power of Secession on behalf of the states as a legal mechanism. ONly a natural right of citizens to revolt against oppressive governments. Why do Libertarians worship the Founders, except for when they contradict politically correct viewpoints?
But we don't even need to take the authority of the author of the document. We need only look at how secession is a contradiction of the entire concept the Founders created and makes national integrity impossible.
We can also look toward the Northwest Ordinance, which stipulated a perpetual union, to which all the states existing agreed.
By this argument, the government could do anything it wanted for national security.
THis is obvious an instance wherein national security would apply. Allowing secession really does threaten the existence of the nation and weakens it abroad. It's absurd for any nation to allow it, and the Founders weren't fools. They did not design a system such that any state could ignore the balance of power at will by threatening secession and set into motion a chain of events that wuld certainly destroy the Republic. Every president has a duty to defend the Constitution and the country from domestic and foreign threats. Rebels who want to run off with a part of the country constitutes a real clearand present danger.
The confederacy didn't like the results of an election, so it blackmailed the country to try to get its way. When it didn't get its way, it subverted the legal democratic process and engaged in armed rebellion and theft of government property. How is that not illegal? Can you imagine what would happen if a country allowed that? Had the Confederacy won and left, we'd have a hostile, totalitarian regime to the South of us.. This is a direct threat. Not only would it be competiting with us on the contentient for our OWN resources, it would quickly balkanize the continent like Europe. ANy state could hold the government hostage with threats similiar to the SOuth. We'd constantly be in conflict over resource rights, under threat of subversion by COnfederate agents trying to steal states away (that was their intention). As it is, the Confederates "claimed" as part of their "new country" two states that did not join them SO much for their claim to care about States' rights.
The Confederacy was an imperiaistic regime that wanted to destroy the United States and extend across the Continent, displacing the United States.
Do you seriously think the founders allowed rampant treason as right? There is no difference between a foreign power that conquers part of the United States, and 11 states leaving at the point of a gun. Both steal territory, investments, etc.
Even so, the secession doesn't even take into consideration the citizens who live in the State. They have no say.
What about federal property inside the States that decide to go rogue? Do you think they are going to allow federal property inside a hostile nation that left in armed rebellion?
The South STOLE US property in leaving, inherently. THey confiscated material, infrastructure, forts, ships, weapons, etc. This is unavoidable unless all parties agree, an all parties did not agree.