Remington actually made rifles in 7x57, called the Remington Model 1934, based on the Model 1917 for export sales as military rifles. The general appearance was similar to the Mauser, but the action was the Model 1917. (Some reports say it was the Model 30 action, but there were no Model 30 or 30S actions - they were all surplus M1917 actions). The venture was generally not very successful, but IIRC some 2500 were sold to Nicaragua.
FWIW, I am aware of the different markings of the U.S. and British Eddystones, but the OP simply said the rifle was an Eddystone - he didn't say what it was marked.