When somebody in the neighborhood carves a Halloween pumpkin and dumps the seeds along the woods for the squirrels...

Definitely a cucurbit of some variety.

The ones in the second photo are much too close together to be able to keep them all, whichever cucurbit they might be. I'd suggest pulling or carefully digging one up. Chances are at this stage the seed coat is still attached to the root, and you might be able to ID the seed, or let us see and we might be able to be more precise than "generic cucurbit".

I'm betting on pumpkin...

fig_insanity Z7a , could you please tell me where exactly I would find a seed coat at this point? I have seen them on baby seedlings that are just starting to poke out of the soil, like they are wearing a little hat at the top, but these plants are big by now. I realize most have to go, we may only keep one, two at most as we are planning to plant various squashes there and other things that will need a lot of space, such as sunflowers.

FWIW, with many seedlings, you don't have to uproot them when thinning, potentially breaking developing roots nearby, you can just snip or pinch them off at soil level.

I agree that they're probably in the squash family. However, my experience is that there are usually crosses in that family-grow a zucchini and it happily crosses with anything else. If you have lots of space you could let them grow and see what come.

Was there a garden there with the previous owners? If not, then pumpkins are most likely as said. before. Unless you have the room to find out what kind of mystery squash for fun, best to pull them and plant what you actually want. Might take up a lot of space.

“grow a zucchini and it happily crosses with anything else...”

Although squash are indeed promiscuous, zucchini, or any other squash type won’t cross with ’anything‘ else. They will only cross within their own species. So, while Zucchini might cross with pumpkin they won’t cross with butternut, for example. More info here https://extension.sdstate.edu/saving-seed-pumpkins-squash-cucumbers-melons-and-gourds 

Fantastic list, floral! but that URL doesn't work for me. Testing to see if this link works:

https://extension.sdstate.edu/saving-seed-pumpkins-squash-cucumbers-melons-and-gourds

eta: My link does work for me. Hover the mouse over your link and I can see extra characters at the end of the link, which explains the difference.

Thanks for updating the link, Party. I’m not at home at the moment and using my ipad on flaky internet.

I have taken another picture of one of the plants that shows bigger leaves now, would this help identifying it ? As curious as I am to find out, I think we should perhaps not take our chances and plant what we really want to grow there ... as it is , we have more seeds than there is space... and we don't need pumpkins.

If you have limited space and lots of seeds to plant: get rid of the volunteers. As the other posters mentioned, curcurbits cross easily with each other. The seedlings you found could be tasteless cucumber or squash things or (if you are lucky) something delicious.