This is a technique I picked up last Autumn from a Japanese TV show (日本人テスト)
Unfortunately, I could only get the gist of what was being explained as I don't speak the language. Even so, it seems to work very well.
In order to cook kabocha (I'm going to try the red Hokkaido squash today) so that it's tender but with the pieces firm and intact without a hint of 'mushiness', here is the method:
You will need a large deep pan and a heat resistant cup (such as a Japanese tea cup).
You will also need deseeded and cut kabocha pieces and roughly the same amount of cooking broth as will fit into the heat resistant cup (a bit more is fine)
broth suggestion: http://justbento.com/handbook/johbisai/frozen-kabocha-squash-plus-easy-s... 
(as an alternative, try adding some lemon zest and a little lemon juice to this broth recipe)
Put the cup upside down in the centre of the empty pan. Scatter cut up pieces of kabocha around the upturned heat resistant cup and pour on the cooking broth. The kabocha pieces should be just covered. Bring the pan to a boil and turn down the heat so it remains at a vigorous simmer.
The upturned cup will rattle a fair bit and can get quite noisy - this is normal. Check on the kabocha every 5 minutes or so (you may want to turn with a spoon occasionally), but by around 15/20 minutes the kabocha should be tender yet with a firm texture (check with a toothpick/skewer) and most of the cooking broth will have disappeared.
Remove the pan from the heat.
Next step is to carefullly lift the cup by an inch or so.
The cup will be full of the cooking broth!
Let the kabocha pieces cool in the broth before removing them.