Yes, there is some evidence that carers are more likely to be ill or disabled themselves, but there are a number of possible reasons for this, and the figures vary a lot according to their situation.
The incidence of ill health and disability are related to poverty: and poverty is also strongly related to educational achievement: so for example well-educated carers with professional qualifications are more likely to figure out how to juggle caring and flexible working partly because they can earn more per hour and are better at finding careers that suit their own caring responsibilities. A GP or teacher who is a carer can easily become a locum or go p/t and earn say £500-£1000 in say 10-20 hours work a week: clearly they are in a far better position to buy in some respite than a carer who is a cleaner and earns £6 an hour.
There is also much evidence that the more income people have, the more exercise they get: whilst walking is free, many other forms of exercise require access to a gym, a bike, or specialised clothing, and all these things cost money.
And when we link this with depression, we also find that poor carers are far more likely to drink heavily and smoke in order to self-medicate.
There is a theory which states that if ever anyone discovers exactly what the Universe is for and why it is here, it will instantly disappear and be replaced by something even more bizarre and inexplicable.
There is another theory which states that this has already happened.