No, unless you have some oily contaminant such as road tar on the siding, do not use mineral spirits. The road salt that you removed was probably white aluminum corrosion (aluminum oxide). After you clean and rinse the surface, you will need to prime all areas of exposed aluminum (that is, where you can see the metal), or where any of the white salts were.
For the primer, use a latex corrosion-inhibitive primer. Ask for this at a paint store. You may find it just as easy to prime all of the siding, rather than just the areas mentioned. If you do prime everything, you will get a more uniform appearance from the paint, compared with if you only prime some parts. For the paint itself, use a top of the line 100% acrylic latex house paint in a flat finish.