I have planted some cucumbers (forgot what species) and they have all sprouted, however, I have noticed that one still has the seed shell/case thing (not sure how to describe it). I am not planning on doing anything to it but it got me wondering, would taking it off actually do anything?
Some people call it a hammerhead. Some will tell you to just leave it and it will come off on it's own and if you try to remove it you risk damaging the plant.
In my experience, if the seed shell is still attached to both cotyledons (the first seed leaves) then it usually doesn't come off on its own. In some cases the cotyledons will grow out anyway but still stay pinched together and growth will be stunted. In other cases so much of the leaves are covered the plant won't grow any more.
If you can remove the seed then the plant will grow normally. You can try to remove it but there's a chance you can accidentally rip of one or both cotyledons or damage the plant in some other way but if you remove it the plant will grow better. At least that's been my experience.
What I do is spray water on the seed and let it soak in about a minute or so. Then I try to pinch the sides of the seed to open it up like you might do with a sunflower seed. Then slowly see if it will pull off gently. If not I might wait and try again the next day. Most of the time I get it off without incident. Cucumber seeds are pretty big so easy to handle but try to get it off before the cotelydons get too big.
Just leave them alone. Nature has had millions of years to deal with them. I've seen the same on my spring onions, left them, and they eventually fall off. If they don't, and the plant suffers, well, that's evolution. And that plant won't be propagating.
I faced that same dilemma with some cucumber seedlings and decided to remove the seed casings. Out of 24 seedlings I damaged one of them. The damage was slight and the seedling did fine.
I was going to leave the seed casings on some of them as a test but could not resist taking them all off. So I'm not sure whether it would have been fine to leave them. I did notice on one runt of a seedling the cotyledon never developed fully and parts of them were still a pale white as though part of the seed.