What is the best toothpaste for zirconia crowns?

Check with your dentist. Whether your teeth are coated in porcelain or completely made of zirconia, is a key factor in knowing how to care for your teeth. Porcelain coated teeth can lose their “shine” and scratch while Zirconia really cannot.
Make sure that you are attending your regular cleanings and check-ups to make sure your teeth are in good condition. 

A zirconia crown won't be damaged if it's eMax, which is typically used in the front, you can abrade the crowns and cause little scratches that collect the stain with your hard brush and whitening toothpaste. Go back to the dentist to get them polished, and then use only a soft brush and non-whitening toothpaste. If you add in some water or soar free gum after having anything that stains, it can decrease the amount of staining that occurs.  

You are correct in needing to know if porcelain is indeed layered on top, or if the crown is entirely zirconia.  A zirc crown is basically "bulletproof" and cannot be damaged by toothpaste, but porcelain on top can lose the "gloss" or "lustre".