With Ekko, you usually want to avoid Juggernauts, especially Garen who I would say is the best anti-assassin of the group thanks to his silence making it difficult for mobile champs to avoid his counter-burst.
I'm assuming you're talking about Ekko top, or Ekko mid vs a fed Juggernaut, so I'll try to break down both situations for you, using Garen.
Garen's early game is heavily overrated, and you should capitalize on that. I don't expect that you'll get many kills unless he overestimates himself or you bring Ignite to the lane, but your E-Q comes out too fast for his silence to interrupt, so you can do your normal combo and speed up to get away. Keep in mind that his Q cleanses slows, so if he uses it after yours he can probably follow you for a second or two and throw damage back. The most important thing is to keep his passive down as much as you can; the regen isn't much early game, but he'll still out-heal you in lane given the opportunity. The best thing to do is to zone him off CS and starve him into the mid game, then roam to get a bigger lead. Since you have mobility+CC, your roaming is much better than his, especially with Teleport.
Against the likes of Darius or Illaoi I would say avoid them altogether. Illaoi's entire playstyle revolves around baiting you into over-committing, and if you get too greedy she will burst you as fast as pre-rework Rengar. Don't fight her anywhere near her tentacles; your E's blink can let you phase through her E if you time it right, which is probably the only way you'll win a trade against her. And Darius is very good at dragging out what you wanted to be short trades; between his hook and his two very strong slows, it's highly unlikely that your attempts at making quick trades will go in your favor, even with your shield.
In teamfights, if you HAVE to deal with them, I would not engage them directly, especially if you've built AP assassin as you are squishy. Focus on helping your team kite them, with your W and Q to stun or slow them. The best thing you can do for your team is to not let them get where they want to be; if Darius or Illaoi manages to get in a good spot to start healing and doing damage to as many people as possible, you're in trouble. Take what damage you can, but be very careful about all-inning them as they are more than capable of turning it around on you.
Juggernauts are almost universally weak to strong poke, kiting, and heavy disengage. % Damage is also good against them as many of them will prioritize large health pools over the more balanced durability that tanks opt into.
