My opinion would be that those baby seedlings are way too young to be outdoors. I don't set out my seedlings until they have several sets of true leaves (those above the initial cotyledons). Even then, they should be introduced gradually---just a few hours in a shady/protected area and gradually increasing exposure. Spinach can take some cooler temps, but these babies are still pretty young.
Did you start these plants under grow lights (they look pretty healthy all in all)? If so, just keep them indoors under some bright light until the weather improves. A few spots on the cotyledons (seed leaves) probably won't hurt. It's not unusual for many plants that the cotyledons turn yellow and fall off once the true leaves start to grow.
Also, if you can bottom water (can't tell what type of growing pots you have), it's better for the baby plants. Don't fertilize more than 1/4 strength once you have at least 2-3 sets of true leaves.
