I tried accessing the secret level (the one you get to by playing the first level three times in a row and clicking the "missing child" poster in front of the house each time) and the third time around the screen flickered as if I was going into the secret level but instead I was brought to a screen asking for a beta license key and the game apparently froze. This has happened both times I've tried.

Is this just a random glitch or an issue with the gog version of the game?

I had this same problem as well. I asked on the Facebook page and they said to ask GoG about it, so I'm pretty sure the GoG version just has a weird bug. Maybe Slenderman is getting revenge for all the negative user reviews on his game.