Can a "Blender Game Engine" powered game be built for Linux?
Yes. Blender Game Engine powered games can run on all major platforms, and support for Android deployment, although limited and regarded as experimental by the community at large, is gaining ground. Deployment to Linux based systems is fully supported, and is a primary focus of the Blender Game Engine.
Keep in mind that a game created with the Blender Game Engine must be built for the system in question by someone with rightful access to the source files. This means that a developer must build their game for Linux in order for the game to run in a Linux environment. The same holds true for the other available platforms. A game built for Windows will only run in a Windows environment.
One exception is Android, where (correct me if I am wrong) support only exists through a native application which requires the source file in
From the Official Documentation for the Blender Game Engine: