Maximize quadratic function subject to linear constraints Can one maximize $\sum_i c_i x_i^2$ where the $c_i$ are constants (possibly negative), subject to linear constraints over the $x_i$?

This paper seems to come close to answering "no."  They show it is NP-hard for target functions $x_1 - x^2_2$.  However they have $x_1$ which is not squared, and from the two pages I can access online I can't understand if that's critical or not.

Bonus question: Is there a free software that can be tried to solve these problems (possibly heuristically)?
It's NP-hard. 

Here's a reduction from the feasibility version of Binary Integer Programming (BIP), which is NP-hard. The problem is to decide if there's a feasible solution to the constraints $Ax \leq b$ and $x_i \in \{0,1\}$. It's easy to convert this to a problem with the constraints $Ax \leq b$ and $x_i \in \{-1,1\}$.

Now consider the following optimization problem: 
$\max \sum_i x_i^2$ subject to the constraints $Ax \leq b$ and $-1 \leq x_i \leq 1$ for all $i$.

This problem has objective value $n$ (the total number of variables $x_i$) if and only if the original BIP problem was feasible.