Do symmetric hydrogen bonds in neutral molecules exist?
It appears that perhaps water may have symmetric hydrogen bonds between oxygen atoms, but it would take at least 60 GPa of pressure to make water to bond like that.

Article named very descriptively "Compression of Ice to 210 gigapascals: Infrared Evidence for a Symmetric Hydrogen-Bonded Phase" proves it:


  Protonated and deuterated ices ($\ce{H2O}$ and $\ce{D2O}$) compressed to a maximum pressure of 210 gigapascals at 85 to 300 kelvin exhibit a phase transition at 60 gigapascals in $\ce{H2O}$ ice (70 gigapascals in $\ce{D2O}$ ice) on the basis of their infrared reflectance spectra determined with synchrotron radiation. The transition is characterized by soft-mode behavior of the $\nu_3$ $\ce{O-H}$ or $\ce{O-D}$ stretch below the transition, followed by a hardening (positive pressure shift) above it. This behavior is interpreted as the transformation of ice phase VII to a structure with symmetric hydrogen bonds.


On the other hand, article "The Hydrogen Bond in the Solid State" mentions:


  First example of an $\ce{O-H-N}$ hydrogen bond with a centered position of
  the proton: pentachlorophenol/4-methylpyridine at 100 K characterized
  by neutron diffraction studies (...)


Unfortunately:


  Even very small chemical changes in the system lead to loss of the
  symmetry (...)


so symmetric hydrogen bonds can probably exist also in very low temperatures.

References

Goncharov, A F; Struzhkin, V V; Somayazulu, M S; Hemley, R J; Mao, H K. Science 273.5272

Steiner, Thomas Angew. Chem. Int. Ed. 41 1 71