I start mine every spring indoors early in 4" pots and have had great success. They are usually doing quite well by the time they go into the garden. Make sure you are starting with disinfected pots and new soil. Then make sure they are getting enough water. Squash use a lot of water. They should never be allowed to get stone hard dry while not sitting in water either.

Are they in a draft after they are moved? I am assuming, since they spourt that it is not too hot on the heated floor for them but perhaps the change to cooler air is a shock. You don't need bottom heat to germinate them indoors if you soak the seeds in warm water for half a day first.

I would try germinating them in 4" pots on the windowsill where they will be growing. Don't move them from their warm floor to the winter window and see if that helps.