The file must be opened by a specific program, the operating system itself does not open it. For example, If you have Illustrator on your Mac you can open it in its original state through Illustrator's File>Open option. You will also be able to place it in InDesign but it will not be editable there. You can open it through Photoshop but you will be asked if you want it rasterized (which means you lose edibility) or opened as what is called a "Smart Object" which will allow you to access and edit the original file by double-clicking on a little icon which will appear in its layer within Photoshop's layers panel.
You should also know, however that .eps is a virtually obsolete format. It is far better to save Illustrator in its native .ai format. Additionally, you should make sure that any type used in the Illustrator file is opentype which will operate correctly on both the Mac and PC.