Many corporations are incorporated in the State of Delaware for Tax and legal reasons. I beleive it has to do with a more favorable business climate.:dancingbanana:
Midvale Steel eventually took over operation of the Eddystone plant from Remington but i believe that part of the story is covered in the Remington Society Article linked in Chuck's post above.
One thing for sure Eddystone was the 800 lb Gorilla in M1917 production. By the end of WW1 Eddystone had made as many M1917's as Remington Ilion and Winchester combined. While Ilion undoubtedly sent some people down to help get Eddystone production set up and then provided highlevel management it's not likely Ilion had sufficient numbers of qualified people just 'hanging around' to go off and run another(twice as large) plant for several years. The skilled workers - toolmakers, machinists, some assemblers would probably have been Baldwin employees.