Yes you can bring water to a boil in an electric skillet but it might take a bit longer for the water to heat up enough to start boiling but it will work.
I've boiled water in an electric skillet before and it did work so yes you can boil water in an electric skillet.
I've cooked spaghetti in an electric skillet before and just filled the electric skillet halfway and then added the spaghetti and covered it with the lid and let it boil.
You can also add a bit of salt to the water and it will help with the spaghetti cooking.
I prefer to use my regular stove now to boil water but when all I had was an electric skillet the electric skillet did boil the water.