Two short remarks related to this problem and Eric's conjectures:
1) In the paper
C. Ballot and F. Luca, ‘ On the equation $x^2 + dx^2 = F_n$ ’, Acta Arith. 127 (2007), 145–155.
the authors show that the equation $x^2+y^2=F_{2n}$ has no solution for most integers $n$.
2) In the problem section of the 42th volume of the Fibonacci Quarterly, 2004, it is shown that the set of integers $n$ such that $F_n$ is divisible by a prime of the form $4k+3$ has asymptotic density $\frac12$.