There are a few things you have to think about, a few risks too.
How big is the cricket? If it's like.. fruit fly sized (About the size of a sesame seed) you can try it, but watch him eat it, and if he has trouble, remove it. If it's larger he runs the risk of choking.
Is this a wild caught insect? Wild insects can bring bacteria and parasites into your fish tank and make the fish sick. I wouldn't feed a betta anything caught in the wild.
Also, something like crickets are not a full time food. He'll still need good pellet food as a regular, daily food source, and things like bloodworms, flightless fruit flies and such should only be a sometimes snack, no more then once a week, if that.
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