Composting rarely kills off all tomato and squash seeds. This is one reason that municipal yard waste programs don’t want you to place your pumpkins in yard waste.
If you don’t want the “volunteer” tomato and squash plants that pop up, they’re easy enough to pull. Sometimes, however, they work to your advantage.
After we have our “smashing pumpkins” parties each year, we always get several volunteers either in the yard, in the compost, or both.
This year, one of our pepper beds was severely damaged by (I think) voles. Recognizing that we weren’t likely to get any peppers I allowed a few pumpkin and tomato volunteers that had sprouted in the bed to grow. We’ve already harvested a few pumpkins with a few more on the (diseased) vine and a mess ‘o cherry tomatoes.
Lemonade. Who says you can’t grow it in New England?