What does R. C. Sproul mean by the 'essence' of God? R. C. Sproul discusses and defends the Trinity in this book, What is the Trinity? He explains that God is one essence, composed of three persons, and that the existence of three persons in one essence does not contradict the law of non-contradiction. Now, what does he mean by 'essence'? Why does he use the term 'essence' instead of 'entity'? Does he use the term 'essence' in the same way he would use it in 'the essence of a squirrel' or 'all the squirrelness about the squirrel that makes up the squirrel'?

I realize that there is a similar question: What do we mean by person?  But this one is more focused on the word, essence, which I believe is a technical term in Christianity, not the word person.
To R. C. Sproul and many other Christians, the word essence is a near-synonym of being or substance in a philosophical sense.  It is the English word that he feels best conveys the meaning of the word that the Greeks used when describing unchangeable, ultimate reality: ousios (the present participle of the Greek verb &quot;to be&quot;).
Sproul adopts a Platonic understanding of essence, as opposed to existence, saying that essence:

is the being element of something, the substance of it.

This he contrasts with existence, &quot;which is the becoming element.&quot;  For Sproul, the key is that the word essence communicates immutability or changeless being.
Now, to your follow-up questions.  Regarding the word entity, we know from the above that Sproul doesn't regard it as close a synonym to ousious as essence.  As to why, we might speculate that in modern English entity tends to add a sense of separateness.  Historically, though both words are rooted in the concept of being, essence enters English earlier and is based on an earlier form of the Latin word esse (to be).
Sproul doesn't employ an example like &quot;squirrelness&quot; in this or his other teaching on the Trinity that I've been able to uncover.  But an analogy like that would likely have some merit, since he regards the essence of something as that which is unchanging and defining.  But the difficulty of understanding the &quot;essence of a squirrel&quot; is obviously magnified when attempting to understand the &quot;essence of God.&quot;
Regarding the Trinity, Sproul contrasts essence with subsistence, with the latter term meaning &quot;to stand under&quot;:

Thus, [subsistence] gets at the idea that while God is one in essence, there are three subsistences, three persons, that stand under the essence. They are part of the essence. All three have the essence of deity. [...]
There are distinctions between [the members of the Godhead], but the distinctions are not essential, not of the essence. They are real, but they do not disturb the essence of deity. The distinctions within the Godhead are, if you will, sub-distinctions within the essence of God.


For more of Sproul's teaching on this topic, see his booklet What Is the Trinity, chapter 4.  He goes into a bit more detail in a video lecture.  He also touches on the doctrine of the Trinity in other works, including Essential Truths of the Christian Faith.
Etymologies of essence and entity are from m-w.com.