Eagle has essentially two different types of CAM processing, a new style introduced after Autodesk took over Eagle, and a legacy style from old versions of Eagle.
Assuming you are using the default template, you will be using the new style. In this case, navigate to the silkscreen top and bottom section in the left side of the screen and click the button in the lower left corner of the layer list. This will let you edit the list and add or remove layers from the job.

If you are instead using the legacy style of CAM processing, for example because you are following a guide you to open the gerb274x.cam job file, you need to double click both of the silk screen sections under legacy in the file list. This will convert them to the new style of CAM job and you can then edit them and follow the same step as above.
However, note that you can also move, rotate, change the font size of, and delete the labels themselves directly inthe board editor, which may be a better method than editing the CAM job. This gives you the flexibility to choose if you want to show some of the component names/values but not others, and you don't have to save a special CAM job for this particular board.
I'm a regular forum user, and not affiliated with Autodesk, but I like contributing to the forums to learn new things about Eagle. If this post answered your question, feel free to mark it as a solution. If something needs clarification, feel free to ask.