I mean, it *probably* works. I think all they did for the SP Edition was remove DRM, and hastily remove multiplayer. Does EYE on Steam even have cloud support? I forget. Well if it does, if you didn't save your files elsewhere, just go DL the Steam version again (roughly 3.5 GB) just to fetch <20KB of save files (HL1, HL2 files, etc), and put them in Documents - streumon - eye - Save if they aren't already there. Then uninstall dat Steam version when you're sure you've retrieved all the save files (also copy them elsewhere, just in case). You might also have to edit the avatarlist or whatever, but probably not.
I'd do it, but I feel like it would permanently break my GoG version of EYE, by writing some "do not work" line in my OS's registry or something.
Besides, I already proved lan multiplayer works (at least to some extent) on the GoG SP Edition of EYE. I've done enough.