Should I include negative examples for Tensorflow object detection API?

If you expect your model to differentiate between "found a figure" and "no figure", then you will almost certainly need to train it on negative examples. Label these as "no image". In the "no image" case, yes, use the entire image as the bounding box; don't suggest that the model recognize anything smaller.

In "no image" cases, you may get a smaller bounding box, but that doesn't matter: in inference, you'll simply ignore whatever box is returned for "no image".

Of course, the critical issue here is to try it out, and see how well it works for you.

In general it's not necessary to explicitly include "negative images". What happens in these detection models is that they use the parts of the image that don't belong to the annotated objects as negatives.