I ride a hybrid on 26" rims currently fitted with hybrid tires. As I am 300lbs+ the rear tire runs permanently deformed/flat and the rims ride over the rubber causing tire pinch. Can anyone recommend the tyre tube combination that I should try to obtain that will stop this happening?
As a heavy rider personally, I don't have many issues with flats.
A normal tire on the high/maximum pressure works fine to avoid pinch flats.
The key is to check tire pressure every time you get on the bike. Even a day will allow a tire to soften 10 psi, and that will allow flats to occur.
Road hazard flats are not avoidable except by avoiding the hazard in the first place. :)
Edit: to be clear, my answer is that it is most likely that your existing tires are not fully inflated, rather than that a different tire is needed.
We don't do product rec here, but some general advice: You want to find the biggest tires you can fit into the bike, and run them at high pressure. The pressure written on the tire sidewall is useless (the maximum pressure depends on the rim and the tire), but in all likelihood you will be close to or exceeding it on many tires. The particular model of tire won't really make a difference if you're pinch flatting them -- that is a symptom of too low pressure.
Also, riding on a flat tire is bad -- you can damage the rim and wheel that way.
If the biggest tire you can put in at the highest pressure you can run safely doesn't work out for you, consider switching to a rigid mountain bike, where you can easily get 2+ inch tires in, which will easily take 300 lbs.
Note that running at a high pressure leaves you vulnerable to other things like road hazards damaging the wheel. You want the lowest pressure such that the rolling resistance is negligible, you avoid pinch flats, the tire absorbs some of the nature of the surface and keeps you in control of riding (i.e. no bouncing). For a 300+ lb rider, this will be a pretty high pressure on many tires. Also, remember tires are supposed to deflect a bit (visibly when you're on the bike), but not too much. If they don't deflect, they're overinflated.
This is close to my goto answer for tyre issues. Tyres designed for touring use are meant for higher loads and inflation pressures. I run marathon plus on my commuter hybrid. They make a 26x2.0 version which is rated to a load of 260lb per tyre and 70psi inflation (which you could probably exceed a little). It's possble that won't fit your rim (see Sheldon Brown) or your frame/mudguards (flip the bike over and pad the tyre out to test it). But you could go a step thinner.
As you've noted, your weight isn't split equally between the tyres - the back wheel carries more load.
You could try converting your tyres to a tubeless system.
I run much lower pressures on my tubeless mountain bike than I used to - they are super resistant to pinch flats. Every now and then I can feel the rear wheel pinching in a way that would previously have caused an instant flat but to date no flats (touch wood!).
I find (anecdotal) that the tubeless tyres seem to lose pressure a little faster than normal inner tubes.
Combine that with regular pressure checks and you should be better off.
Schwalbe Big Ben or Big Apple tires. Maxxis Hookworms.
I'm 450lbs and ride Maxxis Hookworms. Which are 26X2.5 tire. I always ride at maximum PSI. But They might not fit in your forks. I had to buy new forks for the tire to fit.
Another good beefy tire the Schwalbe Big Ben tires. Rated for riders 350lb. Which would most likely fit your bike with no alterations.