Hmmm... I think I would need to know more about their tactics and equipment, as well as how they fit into overall strategy, to make any tenable suggestions. Typically speaking, armies don't just find/make the "best soldier", give them the best equipment, and send them out to randomly have at the enemy. If you are dealing with formation style combat, they could be used when the armies are engaged with one another in a line breaker/shock trooper capacity. If you are going this way, you want a name that suggests strength and toughness. Alternatively, if their tactics take advantage of their speed and agility over their toughness, they could be used in a flanker capacity, either around the sides or over the top if they have jetpacks. For the former, you want something which implies speed and agility, and for the latter, maybe something which implies the air, like a bird of prey or something.
If you are going for more maneuver warfare (formation is more like ancient/Medeval warfare, and maneuver is more like modern warfare), then the role will inevitably get more specialized. Are they used primarily as a fast attack unit, taking and holding ground in initial assaults until the "softer" troops come in? Or are they more for defence after the assault troops take the land? Or, do they function more like mechanized infantry, using their armor to support tanks and other armored vehicles which are typically vulnerable to infantry?
Soldiers rarely specialize in multiple areas, especially if they are valuable and expensive (and these are not necessarily the same thing). So, if you don't already have an idea how combat works in this setting, and what role these units play, I would start there and build out.
Also, are they the only army of the nation which produces them, or are there other units the nation fields. Are they only produced by one nation, or do multiple sides have them? If they only belong to one side, and that is the only sort of army they have, you could just simply call them by the name of the nation they represent.
Anyway, those are my thoughts on the matter. I hope something in there is useful!
