WRONG. The power to suspend is the writ is listed in Article I, prefaced by, 'All legislative Powers herein granted shall be vested in a Congress of the United States, which shall consist of a Senate and House of Representatives.'
'All' as in ALL powers. 'Legislative' powers, not Executive nor Judicial, not shared. 'Shall be vested' in Congress, not the President. 'Herein' granted, as in this Article.
Article II states, 'The executive Power shall be vested in a President'. No mention of suspending the writ. Same for all other Articles.
Chief Justice John Mashall thought so when he wrote, 'If at any time the public safety should require the suspension of the powers vested by this act in the courts of the United States, it is for the legislature to say so. A decision of the US Supreme Court. Just as Chief Justice Taney thought it was for the legislature to decide when he wrote that it was, 'one of those points of constitutional law upon which there is no difference of opinion, and that it was admitted on all hands that the privilege of the writ could not be suspended except by act of Congress.'
Chief Justice Marshall did believe that. And had the matter come before him I have no doubt that he would have voted that only Congress could suspend habeas corpus. Heck, if I was on the Supreme Court I would rule that only Congress can suspend habeas corpus. But I'm not on the court and the matter never came before the court while Justice Marshall was a member, and the Constitution is not clear on the matter since it does not say only Congress can suspend it or that it can be only suspended by an act of legislation. So the matter is still open.
But unilateral secession, on the other hand, was settled by the Supreme Court in 1869. They ruled is was illegal, and so it remains.