Is there a way to automatically burn in clip numbers for an exported file?

You can use the FCP built-in "timecode" effect, which can add both clip name and timecode to all your clips at once. You can't disable the timecode field, but that's often useful anyway. This effect gives a wide choice of font, position, transparency, color, etc. By default, it denotes the timecode base with an on-screen label, e.g, "project", "source", etc, but you can delete that in the effect, then it won't be applied to any clip.

So you can apply a timecode field and a clip name field to one clip or 100 clips in a single step. Note: the timecode effect is different than the timecode "generator."

Also, and just an FYI; Metaburner is a more advanced plugin that does this, available in the App Store or FxFactory. Depends on how deep you want to go. Metaburner is also much more expensive (but you get tons of functionality). I've got both, and almost never touch Metaburner, don't need all of that power. Fox's does the job for me.

If you mean clip names, fox_m here has an effect called Clip Name to Title at fcpxtemplates.

To show the effects pane, type CMD+5. In the search field, type "timecode." For a brief tutorial, go to Youtube and query on "How to burn in Source Timecode in Final Cut Pro".