Answer: Congratulations on your seeding success!
Unfortunately, these plants do grow very quickly. They grow so quickly in fact, that they are usually started much closer to the last expected frost date. They also do not tolerate frost and are transplanted outside at or after the last expected frost date.
If you plant the seeds indoors, you would plant melons at the frost date, cukes about a week before it, and squash about two weeks before it. The plants would be transplanted in about two weeks after seeding. Since they do grow so fast, many gardeners simply direct seed them in the garden, too. Direct seeding would be done about a week after the last frost date when the soil has warmed up a bit.
In my experience, it isn't possible to keep these seedlings healthy under lights for more than about a week or so, so I am afraid you will have to start again later on this spring.
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