This is a tricky question. As stated, the Essential will not officially work with Project Fi. However, it should "work" -- many people have got non-Google phones (even an iPhone) to work with Fi, though there are limitations. First, you must use an officially supported phone to start your service and, second, the phone will likely only use T-Mobile towers, it will never switch to Sprint. Also, you can get a "data SIM" for the Essential, while still using your primary Google phone, and then use Hangouts for talk and text.
As he stated, it is the Nexus 6, 5X, 6P and Pixel phones that will work on Project Fi. The Moto X4 will also work, though I'm fairly sure you have to buy the Android One edition that is only available through Project Fi, that other versions of the X4 will not work.