The primary problem that diving rebreathers solve is eliminating toxic levels of CO2 without throwing out yet-to-be-used oxygen at the same time. As such, it is only useful if your ability to remain submerged, or without an oxygen source, is limited by CO2 buildup, but not by actual oxygen supply--e.g., because you've got a big pressurized tank of the stuff on your back; or because your space station has plenty of oxygen to spare, but you don't just open the airlock and waste it every time you need to cycle out CO2. Given that every deep-diving air-breather I know of solves the problem just by storing up tons of O2, and not worrying about CO2 rejection, this seems like an unlikely situation for a realistic biological creature to find itself in; but we'll assume for the rest of this answer that your creature already has huge stores of myoglobin, and/or other oxygen-carrying abilities, to the point that it really is CO2 buildup limiting its diving endurance.
Contrary to the question comments, a gill is not by itself necessarily a rebreather, since a typical gill will waste oxygen if the external concentration is equal to or above the blood concentration, just like exhaling from a lung. If you are getting your oxygen from the surrounding water anyway, then that doesn't matter, for exactly the same reason that it doesn't matter when you exhale excess oxygen from a lung--you can always just breathe it back in again. But if you can't breathe water, or you are swimming through anoxic water, a gill is worse than nothing; it's like being in space without a spacesuit, and letting all of the oxygen in your blood diffuse the wrong way out of your blood through your lungs (or, for that matter, like being a room filled with pure nitrogen--you can breathe, and it won't even be uncomfortable, but all the oxygen will flow the wrong way, killing you faster than if you just held your breath!)
However, you could imagine a modified gill that selectively passes CO2. One option might be to use an auxiliary circulatory system with a different carrier molecule that picks up CO2 from the primary blood supply, but not oxygen, and transports it to the "rebreather gill". In that case, there would be no need for any special diet, because there would be no consumable resources involved. Now, I've been assuming that this creature is, in fact, a diver, just because that's the most obvious situation that would cause limited access to oxygen, but if for some reason it's a land creature that just happens to frequent anoxic zones, the same set-up could work with an auxiliary lung that selectively passes only CO2.
Alternately, it's certainly plausible to have some sort of special organ/metabolic process that chemically captures CO2, pulling it out of the blood supply without venting it to the environment (analogous to how a mechanical rebreather pulls CO2 out of air). The creature could, for example, build up nodules or carbonates (or oxalates--that saves you one oxygen atom over a carbonate) somewhere, which would presumably have to be expelled, or later decomposed. One fairly straightforward option might be to modify the kidneys and urinary tract to intentionally manufacture and painlessly expel kidney stones. That might require a slightly specialized diet, to ensure that the creature gets enough of whatever minerals are being used to sequester the CO2--likely calcium--but not so much as to be obviously notable. If, rather than expelling the nodules, the creature later decomposes the nodules to exhale the CO2 over a longer period of lower metabolic stress, then no special diet would be needed at all. Additionally, you could imagine a more direct process where the creature, e.g., concentrates and expels carbonic acid directly in liquid form, which again would not come with any special dietary requirements.
Now, since you mention microbial plant life, you may be thinking of a "rebreather" not merely as an cleaning apparatus, but as an oxygen generation / storage device in an of itself. Microbial plant life won't help much. Obviously, they'll only work at all if the creature is illuminated. But even then, their oxygen production rate is not likely to be particularly impressive. There animals that get energy from symbiotic green algae, but they are low-energy things like flatworms. For an active animal, symbiotic algae may technically extend the time it can go without breathing, but not be enough for anyone to care. If stocking up on oxygen when its available and storing in, e.g., myoglobin just isn't good enough, your next option would to go with eating things like chlorates and perchlorates. Chemical oxygen generators can also function on reactions using things like superoxides and peroxides, but those don't tend to occur naturally, have short half-lives, and are really toxic. Superoxides are occasionally produced biologically, so in theory perhaps they could be acquired by predation... but they are produced biologically specifically as poisons in immune response and microbial warfare, and they don't last long, so explaining why there's some other creature that serves as a useful source of them would be an even bigger problem than explaining how this creature works. Chlorate and perchlorate minerals, however, do occur naturally (though typically in hyper-arid regions), and are used as oxidizers in microbial metabolism. So, if you can find an excuse for why this creature has easier access to concentrated deposits of chlorates than it does to oxygen, you could have it eat those, rely on symbiotic microbes to help metabolize them, and excrete the excess chloride ions. Nitrates and sulfates could also serve a similar role as raw materials for an oxygen-generation metabolism, or simply as alternative oxidizers directly, and you might have an easier time explaining the existence of convenient plants / prey animals providing those in usefully high concentrations.
