Deinonychus eggs can only be gotten like the rock Drake or wyverns, you just steal a fertilized egg from their nest. Normally I would use a snow owl to freeze it while my tribe mate grabs the eggs. Or you can have a gun and a dinosaur with good melee. (The gun to shoot it if it latches on your mount). And kill them and grab the eggs.
You can't tame these guys by knocking them out. Instead, you need to steal their fertilised eggs (just like wyverns and rock drakes), hatch them and raise them, just a bit harder than knocking them out. The eggs can, of course, be found in their nests and do be careful while trying to steal one as the Deinonychus is very territorial. Hope this helps :)
Use any flying mount and fly over the nest land and snatch the egg making sure there are no Dino’s nearby and fly away. We got four eggs very early in the game but are stuck on hatching the eggs.. hope this helps. ;)
Here's how to steal one early game: If you spawn in at chalk hills 3 and spawn next to a pond walk opposite to the pond. There are some nests right there. Snatch one quickly and run off the cliff next to the pond. Be careful and make sure to fall into the water. From there make a left and you should get to a beach area. From there it should be easy to find your base.
1. Tame a kangaroos 2. Have a tribe mate with a whip in the pouch 3. Quickly jump on a nest and have your tribe mate whip the egg into his inventory 4. Jump away! No one has to dismount so there’s basically no risk