Had a bit play with it earlier, and like most of the nrich stuff, it's well put together. I suppose that there isn't a simple winning strategy (it appears that the computer's response has some element of a random response), although obviously, there is a symmetry to start off with which cuts down the possibilities a bit.
I was losing my initial attempts where the computer won with squares which had sides length (root(3), root(4), ... i.e. they went along a couple and then up/down one). However, a bit of concentration was needed to stop this.
One of the key patterns to stop / try and get is three isolated like
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Then you're into a winning / losing end game as the opponent must counter with one to stop the obvious rotated square,
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then you can put one in one of the other corners and there are two squares which could be formed, hence win / lose.
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So a simple strategy is to form / stop a "<" type pattern being formed without any opposing counters around it.