You will recover some egg shells from the Jersey Devil's nest in the forest.
You will need to go to the other 3 locations first. You need the matches from the Pub at Loch Ness, the lantern from the monastery at the Himalayas, and the bolt cutters from Puerto Rico (at the farm that is the upper left star on the Jeep map).
At New Jersey
Open the dumpster with the bolt cutters. Take the stall door into the restroom and put it with the one that is still there, giving you driving instructions for the dark maze of the forest. Use the matches to light the lantern. Drive according to the directions to the old house in the forest (you can get there faster by turning right, left, right, left, right, and left). Climb to the attic to get the Grappling Hook. As you see the Jersey Devil, follow it outside and use the grappling hook to climb the tall tree. Get the egg shells and take them back to the Mews lab. The DNA test will show that the Jersey Devil (an unknown animal) actually exists.
Take the shells back to the Mews lab and test them. The DNA analyzer will prove the existence of the Jersey Devil.
There are 3 you have to prove: Nessie (Loch Ness) Chupacabra (Puerto Rico) New Jersey Devil (New Jersey) You can't get what you need to prove the Yeti, so you are sent to find Bigfoot.
You still have to prove the chupacabra and the Jersey Devil. You can get a photo from the Himalayas, but it does not prove the Yeti. So when you have the other three proven, you will be sent to chase Bigfoot in the Northwest.
It's not the chicken eggs, it's the egg shells from New Jersey. When run through the DNA testing, the egg shells from the nest in New Jersey prove the existence of the Jersey Devil.
After you have the chupacabra tooth, you still need to prove that Nessie and the Jersey Devil exist. (The monks are stubborn about the Yeti and your snow footprint is unconvincing.) Get the photo of Nessie, and some egg shells from the Jersey Devil, and then you can pursue Bigfoot instead of the Yeti. (see related questions)
You cannot prove the Yeti based on the evidence from the Himalayas. So you have to prove the other 3 (Nessie, Chupacabra, and New Jersey Devil) and when you do, Mews will send you to pursue Bigfoot in the Pacific Northwest.
Prove 4 cryptids to qualify for the Mews million-dollar reward. You find a footprint that may be from the Yeti, but you can only prove Nessie, the Chupacabra, and the New Jersey Devil. Then you have to track Bigfoot and save him from Gretchen Grimlock. (see the related solution question)
Run the DNA test on the eggshells to prove the existence of the Jersey Devil. You will also need to prove Nessie and the chupacabra, and bring back the photo of the footprint from the top of the mountain in the Himalayas. When all but the Yeti are proven (Nessie, chupacabra, and the Jersey Devil), Mews will send you after the Bigfoot in the Pacific Northwest.
Once you have gotten the lantern and the footprint photo from the Himalayas, you still cannot prove the Yeti. You can prove the other three cryptids (Nessie, the chupacabra, the Jersey Devil). Once you do, a fifth location will appear on the helicopter controls: you must go to the Northwest to chase Bigfoot.
You cannot prove the Yeti. But after you confirm the other three cryptids (Nessie, Chupacabra, and Jersey Devil), Mews will send you in pursuit of the similar creature, Bigfoot, in the Pacific Northwest. The location will be added to the GPS on the helicopter.
After you have confirmed the other three cryptids (Nessie, Chupacabra, and the Jersey Devil), and brought back the inconclusive footprint from the Himalayas, Mews will send you in pursuit of Bigfoot, which is similar to the Yeti. Since you have him in a cage at the end (but let him out), he is definitely confirmed.
You cannot prove the Yeti based on the evidence from the Himalayas. So you have to prove the other 3 (Nessie, Chupacabra, and New Jersey Devil) and when you do, Mews will send you to pursue Bigfoot in the Pacific Northwest.