Answer: Many people have had relatively late tomatoes and morning glories and peppers -- and eggplants -- due to the cold cloudy slow spring and late start to the season. I think too that the heat spell may have caused some fruiting issues with pollination problems. If your plant is flowering you should eventually see some eggplants on it. However in your zone 5A location, it is possible that your season is just too short to get a good crop of eggplants. They like warm soil, warm sunny weather and then need a long season to mature anbd ripen.
Based on your description I am not sure what is causing the spots on the squash; it might be superficial skin damage due to something like hail or careless hoeing, or it might be an early stage of a fungal infection -- these seem to appear especially when temperatures are warm and the weather is so humid for an extended period. If the spots enlarge and begin to soften or rot then that is probably what is causing it. If this is the case, remove any affected fruits from the garden and hope that stops the spread of the problem. You can also try mulching beneath the plants with a layer of clean straw or similar clean dry material; this will help keep them clean and may prevent the spored from splashing back up onto the plants. However there are also some virus problems with cucurbits that can cause corky bumps on the fruit. Your local county extension may be able to give you a more specific diagnosis and if a control is needed will have the most up to date recommendations on what to use and how/when is best to apply it.
You can certainly cut around or cut out the spots when you prepare the squash and eat the healthy parts.
This fall make sure you clean up all of the squash plant debris and remove it from the garden, then rotate your plantings next summer so that you are not replanting a cucurbit in the same spot again for several years. This can help keep any pest or disease problem from recurring. ENjoy your squash!
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