Can I save my weighted companion cube in room 17?

There is indeed a way to save your loyal Companion Cube.

You merely need to take advantage of two deficiencies in Aperture Science's Test Chambers. One flaw in Test Chamber 16 lets you push a Weighted Storage Cube through the Material Emancipation Grill and take it up in the elevator to Test Chamber 17. In that chamber, another defect allows you to use the lesser cube to obtain a second Companion Cube. You can then euthanize it instead of your own faithful companion, thereby tricking the Emergency Intelligence Incinerator and shattering another test subject's dreams for the future.

This YouTube video has the details:

No, I'm afraid not. There's no way to take the companion cube outside this testing chamber, as the door will not open until you've dropped it in the incinerator.

Any stories you may hear about methods of circumventing this process are merely the fictions of devious minds, designed solely for the purpose of instilling false hope. It'll be better for everyone once you accept the inevitability of your hideous betrayal.

The video is propaganda advanced by dangerously bad test-subjects. It is clearly a violation of both the constraints and the intent of the test to employ exploits of this sort, and is frankly bad science.