Why is the following RSA PRNG cryptographically secure? One requirement states that the generator has to withstand the next-bit test.

Consider the following PRNG, where we calculate next output $x_i$ via the formula
$x_i = x_{i-1}^ e\mod n$.

I can see that this PRNG withstands the "state compromise extensions" (can't guess previous states).  However, given $x_i, e, n$ an adversary can easily compute all future numbers.

Why is this generator still considered cryptographically secure?
Perhaps you are thinking of the Micali-Schnorr PRNG, as described in Algorithm 5.37 of the Handbook of Applied Cryptography?
Algorithm 5.37 in HAC never states that $e$ is known to the adversary, or even that $n$ is known. Also, Algorithm 5.37 outputs only the least significant bits of the number, on each iteration. So I think you are confusing RSA as typically used and RSA as used in the Micali-Schnorr PRNG. What you describe does not match what is found in Algorithm 5.37.  Yes, for the generator you describe, given $x_i$, $e$ and $n$, the adversary could compute $x_{i+1}$, but in the Micali-Schnorr PRNG, it appears that we assume that $e$ (and even $n$) is unknown to the adversary.

Note 5.39 states that the security of this PRNG is based on stronger assumptions than just the intractability of the RSA problem. In general, this is not a good thing. In crypto we generally want as weak of assumptions as possible.