The calculus (stones) are called sialoliths, and are primarily calcium phosphate. It is the alkaline nature of the saliva, along with the high concentration of calcium and phosphate that lead to their formation.
This is not going to be easy. If they are small enough they can be squeezed out.
There is only one other thing I can think of, but you are going to have to be careful doing this. Vinegar or citric acid can dissolve the calcium phosphate and any calcium carbonate that may be present in smaller amounts. But, you don't want the acid in contact with your teeth for very long. The second problem is that with the blocked duct you have to be careful about stimulating saliva release as it can create more pressure. So the best way I can think to do this is to take some distilled vinegar, so minerals are removed. Dip a cotton swab in the vinegar then hold the swab on the gland. After about 10 to 15 seconds dip the swab in the vinegar again to restore the acidity and reapply. Keep doing this for a while until it feels like the stone is completely dissolved.