It is a folklore conjecture that for $y\le x$ one has $$ \pi(x+y) -\pi(x) = \int_{x}^{x+y} \frac{dt}{\log t} + O(y^{\frac 12} x^{\epsilon}). $$ This is only relevant for $y \ge x^{\epsilon}$, and is stronger than RH. In the case of primes in progressions, such a conjecture may be attributed to Montgomery. Probabilistic considerations might suggest a stronger error term like $O(y^{\frac 12} \log x)$, but this is known to be false thanks to the work of Maier. But the conjecture as stated above is widely believed. See, The distribution of prime numbers for example for a discussion of this and related work (in particular the discussion around (3.7) there). See also Montgomery and Soundararajan where refined asymptotics for moments of primes in short intervals are made and justified heuristically; these conjectures state that primes in short intervals have an appropriate Gaussian distribution, and imply the conjecture above.
Of course we are very far from the conjecture mentioned above, but if true it implies for large $n$ that $$ \pi((n+1)^2) - \pi(n^2) = \int_{n^2}^{(n+1)^2} \frac{dt}{\log t} +O(n^{\frac 12+\epsilon}) = \frac{n}{\log n} + O(n^{\frac 12+\epsilon}), $$ since $\log t = 2\log n + O(1/n)$ throughout the interval $n^2 \le t\le (n+1)^2$. Now we know that $\pi(n)$ is asymptoically $\text{li}(n)$ which has the asymptotic expansion $n/\log n + n/(\log n)^2 +\ldots$. The secondary term of $n/(\log n)^2$ dominates the error term in the conjectured asymptotic, and so one should certainly expect that for large $n$ one has $$ \pi((n+1)^2) - \pi(n^2) \le \pi(n). $$
An analogous question formulated for $\pi((n+1)^3) - \pi(n^3)$ is known, as mentioned by GH from MO in the comments, by Huxley's version of the prime number theorem in short intervals.
Now the question is only asking for an upper bound on $\pi((n+1)^2) -\pi(n^2)$, and one does have unconditional upper bounds by sieves. The Brun-Titchmarsh theorem would give a bound like $4\pi (n)$ for this quantity, and one can do somewhat better than this. The best result that I know is due to Iwaniec from whose work (see Theorem 14 there) it follows that $$ \pi((n+1)^2) - \pi(n^2) \le \Big( \frac{36}{11}+ o(1)\Big) \frac{n}{\log n}. $$