Hail Bagpuss!First, follow Crane Driver's advice. Sit down, place one end of the box on your knee/thigh and play from that position. Most players I know set the left end on the left knee and let the right hand do the pushing and pulling. This will relieve your hands from the weight. Morris players often use a separate set of wrist straps to make it less strenuous to play while standing. You should also go to www.concertina.net. One of the regular posters whose name has slipped from my mind has some suggestions as to rater unorthodox ways to support concertinas -- but they do seem to be good ideas.
Which toe is broken? If the right, rest the concertina on that knee' you can't play and keep time on the same foot unless you are looking for a wha-wha effect which isn't particularly pleasant.
It is also possible that ypu are playing for too long a stretch. Pace yourslef. Play for only as long as it is comfortable and enjoyable. Ten stretches of fifteen minutes will yield more results than a long session of two hours or more.
I've only been playing English for twenty five years, but the only times my hands get stiff is after a medley of Road to Listoonvarna?Merrily Kiss the Quaker's Wife/Blarney Pigrim too fast.
Keep on squeezing -- Tom