Any luck? I'd be interested in seeing some of the computer solutions even if their scores didn't beat mine.
By the way I can now improve my score to 14sqrt(3)-24 = 0.249... . My covering shape is a 1/4 by 1/7 right-angled triangle. This clearly tiles the square perfectly and you can also fit 24 of them into the equilateral triangle. To see this first divide the equilateral triangle exactly into 24 right-angled triangles of sides 1/4 and 1/(4sqrt(3)), and then note that 1/7 < 1/(4sqrt(3)). There's no point in drawing a picture since you can barely see the gaps.
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