It wouldn't work out that well if you just had heat tape and the shoe box tub.
It wouldn't heat the eggs evenly and might not even heat the eggs at all. I really doubt it'll be able to penetrate through the substrate medium without it exceeding a dangerous temperature. And at that point, it'll be more prone to error and malfunction.
If you have a cooler lying around, you can use that to insulate everything.
Secure the heat tape to the side wall in the cooler. Set water bottles on their side on the bottom of the cooler.
Lay down a light grate/diffuser over the water bottles to create a solid surface to lay your egg tub.
Connect your heat tape to your thermostat.
Depending on how large the cooler is, you may want to put in a small computer fan to circulate the eggs.
Check out the DIY Subforum here. They have DIY Stickies on how to build your own incubator. (1 from a fridge and 1 from a cooler)
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