For full Hesperornis taming details, get the taming calculator app.
To tame a Hesperornis, you must kill a fish, and drag the fish's corpse to the Hesperornis to feed it. The fish's size plays a major factor in how many fish will be required (a 1.0x fish appears to fill up the taming bar twice as much as a 0.5x fish). Level of fish does not appear to impact taming. The Sabertooth Salmon is by far the most effective fish, equivalent of about 4 Coels of the same size, while the Piranha is equivalent to about 2.4 Coels.
Tame to this kill fish and drag there body to it and once tamed if you feed them more dead fish corpses it will build up and lay a golden egg which is used as a buff to certain dinos or oil or a player buff
Tamed Hesperornis WILL lay golden eggs, just takes a bit of time. I had mine on follow in the Redwoods river, and whistle attack targeted it on a fish. It must kill and harvest SEVERAL fish before going on land to lay an egg. It will either lay a normal duck egg, or it will lay the golden egg. It seems to be random, however mine laid about five normal eggs for every golden egg. If the type and size of fish matters, I have not figured that out yet. It will usually kill all types in one shot, and it will not agro every single sabertooth Salmon in the vicinity when it kills one so don't worry about them ganging up on your duck. I was using a lvl 80, probably tamed around level 50 or so and all I put points in was health. Have it kill every single fish in the river before you give up trying for that golden egg. It will happen, give it time. Hope this helps :)