No, this is not true. -- For example $p := 2^{2203}-1$ is a Mersenne prime (cf. http://en.wikipedia.org/wiki/Mersenne_prime), but $p^2+1$ is divisible by $5^2 = 25$.
Edit: To answer D. Burde's question: $p := 2^{11213}-1$ is a Mersenne prime as well, and $p^2+1$ is divisible by $13^2 = 169$.