As a long as a tree's roots are still viable, it will continue to produce suckers to try to regenerate itself. For your crepe myrtle stump, the only options to stop the suckering are either to grind away the stump as much as possible and cover the area with mulch to allow whatever remains to rot or to apply some kind of chemical stump and root killer over freshly cut sucker surfaces to kill any remaining root system.

For the two youpon hollies, and other similar small to medium trees, suckering is a natural response of the tree and suckers should be removed when pruning. Excessive suckers, though, can be a sign of stress for that particular tree, perhaps caused by damage to surface roots, which may happen if the tree is in a bed with other nearby plants. One suggestion is to tear, rather than cut away, the shoots so that the majority of dormant basal buds are removed, to reduce the possibility of regrowth.