"JSON: c:/cygwin/home/doctoxic/Cataclysm-DDA/gfx/DeonTileset/tile_config.json & /deontiles.png"
There’s your problem. It does not accept the &/deontiles.png part of the string. Remove that and it should work
Did you do anything special to install support for SDL on cygwin? As best as I can tell cygwin takes some extra work to properly support SDL (and therefore tiles) since it’s designed to emulate linux when possible.
A check shows that currently Cygwin and SDL 1.2 (which is what we use) are currently incompatible and therefore it is impossible to compile a tiles version using Cygwin. If we update to SDL2 at a point in the future (which hopefully shouldn’t be too difficult) then Cygwin and tiles builds should become compatible once again. I’ve added a note to the wiki compiling page, hopefully that should prevent people from running into this again.
For now your options are to either download a different program to compile, or just to go without tiles sadly. (Also remember to do “make TILES=1” and then “./cataclysm-tiles” if you want to make and play a version with tiles.)