The origin of the phrase is credited to Gen. Jacques Duchesne, who commanded the French invasion of Madagascar in the 1895. Although the French triumphed in the end, it was only after an absolutely miserable, disease ridden campaign which saw thousands die from disease - ⅓ of the entire force - and only a handful from actual combat. Duchesne himself deserves a decent amount of the blame for his lack of care, pushing ahead on a poorly chosen route based on a poorly decided plan, and Douglas Porch wrly sums it up that "Duchesne's iron will proved more durable than the bodies of his soldiers".

What misery, these evacuation convoys of the sick. They consisted of the food supplies carts that went back empy. Three or four men were packed in there, and the Kabyle driver departed. There were seven more or less to the trip under the sun, with the bumps of the vehicle without suspension, on a pothole-riddled and almost unusable road. What suffering did these moribund endure, lying next to cadavers or sick bodies with dysentery, shaken in their own dejections.

800 men of the Legion were part of the campaign, and suffered as much as any, but also carried it without undue complaint, complimented by Duchesne at the end of the ordeal that "it is assuredly due to you, gentlemen, that we are here. If I ever have the honor to command another expedition, I will want to have at least one battalion of the Foreign Legion with me", although it should be said that the complement was not without a tinge of implicit racism, choosing to compliment the white troops after the mostly black troops that had suffered, and carried through, as well. It reflected as much the unearned trust that French officers implicitly placed in white troops as any actual deserved praise.

Anyways though, the main point is that Gen. Duchesne is credited with that phrase, that the men could 'March or Die', as they most assuredly would if abandoned in the wilds of Madagascar. It was not directed at the Legion specifically, but it certainly is a phrase that they made their own, not dissimilar to that refrain of the US Marine Corps, "Come on, you sons of bitches, do you want to live forever?" it is prideful ownership of their own mortality. It was, truly a miserable experience, but one that afterwards is looked back on fondly... perhaps most fondly by the ones who didn't participate though. but in any case, one lurid description from a surviving participant recounted the assembly of the remaining force just before their assault on Antananarivo:

We were silent, not a shout, not a whisper, not a breath, and yet, fifteen hundred men were lined up. they were, in truth, so gaunt, so depressed, that you would have thought they were dead men rather than alive. Clothes reduced to rages, shoes to pieces, their helmets, too wide for their thinned out faces, fell on their shoulders, covering almost all of those yellow faces, where only eyes animated by fever expressed some life.

It kept with the Legion in part, again, because the bleak romanticism of it appealed, but also because it was their reality. As Gilbert notes:

The ability to march long distances with full kit has always been central to the Legion’s tactical philosophy. Hard marching also had a special importance for colonial troops, and the famous Legion exhortation ‘March or die!’ simply reflected one of the brutal truths of campaigning in Africa and Asia: those men who fell out were liable to be hacked to death by native tribesmen.

Sources

Adrian D. Gilbert. Voices of the Foreign Legion. Skyhorse Publishing, Jan 2011.

Douglas Porch. French Foreign Legion. Skyhorse Publishing, Nov 2010.

Jean-Vincent Blanchard. At the Edge of the World: The Heroic Century of the French Foreign Legion. Bloomsbury Publishing USA, 2017.