In Introduction to Mathematical Statistics by Hogg and Craig, they state the generalization of Chebyshev's Inequality as:
Let $u(X)$ be a nonnegative function of the random variable $X$. If $E[u(X)]$ exists, then, for every positive constant $c$, $\displaystyle Pr[u(X) \geq c] \leq \frac{E[u(X)]}{c}$.
Why do they state this for $u(X)$ instead of just $X$? Is it just a matter of preference for applying the theorem later, or is it actually more general for some reason? It seems like you can just make the substitution $Y=u(X)$.
As mentioned in the comments, it is no more general to state it with $u(X)$ where $u$ is a non-negative function, or for $X$ where $X$ is assumed to be non-negative. One way to see this is that the family of all random variables expressible as $u(X)$ with the above conditions is seen to coincide with the family of all non-negative random variables. (It is easy to see this by using the fact that all random variables can be defined on the sample space $[0,1]$ by applying a measurable function to a uniform random variable.)
Now for the (psychological) question of why an author would state the inequality one way or the other: it is in order to ease the application later. There are 3 ways that this "generalized" Chebyshev inequality is used: for the "first moment method" (Markov's inequality) with $u(X)=|X|$, for the "second moment method" (Chevyshev's inequality) with $u(X)=X^2$, and for proving concentration inequalities by taking $u(X)=e^{tX}$ and optimizing over $t$ in the resulting bound (like proving Hoeffding/Azuma/Bernstein/etc concentration inequalities). In all three cases, one is proving an inequality for the random variable $X$, but along the way one is introducing an auxiliary random variable $u(X)$ and applying the basic inequality to that random variable. Now the "non-trivial" part of the argument is cooking up a suitable function $u(\cdot)$, so it makes sense to emphasize it.