If it is the small squash that soften and drop off they are probably being waterd too much.They will grow large lush leaves but poor fruit, A small courgette on an over watered vine will often turn yellow and rot as it reaches thumb size.
Courgettes also get blossom end rot the same as tomatoes, It starts on what looks like a good healthy courgette. Look at the blossom end for first signs, it will be a dark, possibly sunken spot which will gradually increase till eventually half the courgettes is rotten at the blossom end. This howevr usually is on the first fruit and often corrects itself as the season progresses, it is not infectious.
You can use the good end of the courgette by the way if it is mature.
XX Jeannine