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There would be no way for us to know the specific specs of your particular reed switch, but here is a link to a number of them that are 10mm or less which might work. They are arranged from most to least sensitive.
All of these can handle at least 350mA switching current, which is almost certainly more than enough for your application. Some are more sensitive than others, so the sensing range would be dependent upon the strength of your magnets and the distance between the magnet and the switch. Depending on distance between adjacent squares and the strength of the magnets, it is also possible that a piece on an adjacent square could trigger one of these if it were too sensitive. You may have to experiment a little there.
I would start by finding those which look most similar, dimensionally, and then either select one somewhere in the middle of the sensitivity range (either the 15 ~ 20AT or 20 ~ 25AT range) or get a few of them with different sensitivity ranges to see which worked best.