To reach Duriel in Diablo 4, you need to go through a whole gauntlet of end-game bosses and item grinding, which includes the Mucus-Slick Egg. While you may not be able to guarantee the item every time, I can help you locate the egg and get the ritual started.
Where to Get Mucus-Slick Eggs in Diablo 4
You can get a Mucus-Slick Egg by defeating the Tormented Echo of Varshan in the Malignant Burrow found in the Writhing Mire of Hawezar. This is the same exact location as the Tree of Whispers. They are so close, in fact, that you can find the Malignant Burrow directly to the left of the tree. So you can easily take a waypoint there to start your fight.
There are two different versions of the Varshan fight in Diablo 4. The World Tier 3 variant only requires three items and the World Tier 4 Tormented version requires four of them. In both versions, you need a Blackened Femur, a Gurlging Head, and a Trembling Hand. For the World Tier 4 version you will need a Malignant Heart as well.
All of these items are earned in two ways. Either you get them from opening up Whisper Caches as rewards for Grim Favors, or you can kill Grotesque Debtors. The Grotesque enemies only really appear in Blood Harvest events at random times or at the end of a World Boss fight. Either way, you should use both methods to stock up on items.
Once you have at least one of each item in World Tier 4, head to the Malignant Burrow and summon Varshan with the Malignant Altar. The items will be consumed and you have a high chance of getting one Mucus-Slick Egg if you defeat the boss. Just be ready for a battle that’s at least level 75 in difficulty.
For a better idea of how to get the summoning items, here is my guide on how to get a Malignant Heart in Diablo 4 Season 2.