Can you paint cedar shake shingle siding?

Yes.

From the following site: Tips, solutions, answers and how tos for paint contractors by Sherwin-Williams

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Cedar

Cedar shake or siding, as well as redwood, is best primed with an exterior oil-based wood primer (some regions must use a latex primer due to VOC regulations). A primer is essential – it will serve as a barrier coat to help prevent tannin bleed, a brownish or tan discoloration that can appear on a painted wooden surface. Before coating, make sure that new wood is dry enough to absorb sprinkled water before coating. Top the siding with one coat of Duration.

If you plan to stain the cedar, apply two coats of Sherwin-Williams WoodScapes Solid Color Acrylic Exterior House Stain. This waterborne product is easier to apply than alkyd or conventional latex stains – with no drips or runs. Available in acrylic solid colors or polyurethane semi-transparent shades, WoodScapes resists mildew and retains its color for several years.

One note of caution: Be careful if you decide to go with a semi-transparent stain because the shade of the substrate contributes to the final color, and the finished job may not look quite like what you had planned. It's a good idea to test the stain in an inconspicuous place before applying en masse to make sure it will produce the color you want.

Another bit of advice: An acrylic solid color stain will offer best results for heavily knotted wood, as semi-transparent stains will not hide the knots completely.

If the cedar has never been painted or stained but has been exposed to the elements for more than 60 days, evaluate the level of decay before proceeding. Any loose, fibery, gray-looking areas should be lightly abraded prior to coating.

If the original paint is intact, cedar siding can be repainted with one coat of Duration.

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Yes , you can paint it, but it will hide the true beauty of them, but if they are an eyesore of such already , then yes paint them.