There are at least two distinct senses in which consciousness can be binary. The first sense is the kind you are probably thinking about: the range between e.g. insects, dogs, and humans, or maybe between early and late-stage Alzheimers.
The second sense is the kind that your interlocutors are (I surmise) thinking about. Imagine this: A being that is functionally exactly like you, and that is experiencing exactly what you are experiencing, except that it is experiencing everything "only half as much." It still behaves the same way as you, and it still thinks the same way as you; it's just that it's thoughts only count half.
If this sounds ridiculous to you, well, then you agree with your interlocutors. :) Personally, I think that there IS such a thing as partial consciousness in the sense described above, and I can link you to literature if you like.
EDIT: The place to start is Nick Bostrom's "Quantity of Experience: Brain Duplication and Degrees of Consciousness," available for free online.
EDIT: But the people who ask "Do animals have consciousness" are probably talking about the first kind, in which case I share your frustration. The second kind is more what people talk about when they ask e.g. "Could a machine have consciousness?"
Imagine this: A being that is functionally exactly like you, and that is experiencing exactly what you are experiencing, except that it is experiencing everything "only half as much."
IOW halfway between a person and a p-zombie?
This paper, or more often the New Scientist's exposition of it is being discussed online and is rather topical here. In a nutshell, stimulating one small but central area of the brain reversibly rendered one epilepsia patient unconscious without disrupting wakefulness. Impressively, this phenomenon has apparently been hypothesized before, just never tested (because it's hard and usually unethical). A quote from the New Scientist article (emphasis mine):
One electrode was positioned next to the claustrum, an area that had never been stimulated before.
When the team zapped the area with high frequency electrical impulses, the woman lost consciousness. She stopped reading and stared blankly into space, she didn't respond to auditory or visual commands and her breathing slowed. As soon as the stimulation stopped, she immediately regained consciousness with no memory of the event. The same thing happened every time the area was stimulated during two days of experiments (Epilepsy and Behavior, doi.org/tgn).
To confirm that they were affecting the woman's consciousness rather than just her ability to speak or move, the team asked her to repeat the word "house" or snap her fingers before the stimulation began. If the stimulation was disrupting a brain region responsible for movement or language she would have stopped moving or talking almost immediately. Instead, she gradually spoke more quietly or moved less and less until she drifted into unconsciousness. Since there was no sign of epileptic brain activity during or after the stimulation, the team is sure that it wasn't a side effect of a seizure.
If confirmed, this hints at several interesting points. For example, a complex enough brain is not sufficient for consciousness, a sort-of command and control structure is required, as well, even if relatively small. A low-consciousness state of late-stage dementia sufferers might be due to the damage specifically to the claustrum area, not just the overall brain deterioration. The researchers speculates that stimulating the area in vegetative-state patients might help "push them out of this state". From an AI research perspective, understanding the difference between wakefulness and consciousness might be interesting, too.