I need a real, symmetric window function $x(t) = x(-t)$ whose Fourier transform $\hat{x}(\omega)$ (also real and symmetric) is non-negative $\hat{x}(\omega) \ge 0$ for all $\omega$. The function does not need to have finite support or be causal. An equivalent problem is to find a good low pass filter with a positive impulse response.
A rectangular window would not satisfy this requirement, since its transform is a sinc function.
A Gaussian window $x(t) = \exp(-at^2)$ would satisfy this requirement, since the transform of a Gaussian is a Gaussian, which is positive. However, a Gaussian function does not make a good window because it is too narrow i.e. the cut-off point is too close to the origin compared to the function's decay. Other functions with positive transforms include the Laplacian density function $x(t) = \exp(-a|t|)$ and the triangular function, but these are worse than the Gaussian.
The Hann window (one period of a raised cosine) does not have a positive transform. It can be written $(1+\cos(2 \pi t)) \operatorname{rect}(t)$. The constant and cosine terms have Dirac delta functions as their transforms, hence the transform of the Hann window function is a sum of translated sinc functions.
What is a good window function that satisfies this constraint?
You could experiment by taking an arbitrary real-valued (window) function $w(t)$ and define $x(t)$ by
$$x(t)=w(t)\star w(-t)=\int_{-\infty}^{\infty}w(\tau)w(\tau-t)\,d\tau\tag{1}$$
Obviously, $x(t)$ is real-valued and symmetric. Furthermore, if $W(\omega)$ is the Fourier transform of $w(t)$, then the Fourier transform of $x(t)$ is given by
$$X(\omega)=|W(\omega)|^2\ge 0\tag{2}$$