You completely could. Some hard core cider makers have there own apple presses and make their own juice. The only thing is that most ciders are made up of a combination of apples that create its certain flavor. you can find apple combinations online, but sometimes cider apples are hard to find in a grocery store. I would reccomend hitting up the nearest farmers market around you to find good apples, and good variety.
Primary: blueberry apple cider
Secondary: panty dropper cranberry apple cider
bottled: I should probably get some of those...
on deck: future brews suspended for the moment :(...