All oils are noncompressible, which is why you can use just about anything in a hydraulic system...even water, which of course you wouldn't, number one because it has too low a viscosity and number two it would rust out all the innards. ;-) Of most importance is the viscosity and that has more to do with operating temperature than anything else for a press. In New Mexico, I doubt that cold termperatures are going to be much of a concern for you, so a good medium viscosity oil would serve you well. Other considerations beyond viscosity would be detergent oil or nondetergent. Some hydraulic fluids also contain antiwear additives. I don't think you'd need to be too concerned with detergents nor antiwear in a home press.
If it were me, I'd go up to the local autoparts store and just buy a few quarts of ATF (which is a hydraulic fluid)...the press will work just fine.