Ask Mr. Smarty Plants

Absolutely. Especially if you have berries on your shrubs, the friendly birds who love wax myrtle berries are probably assisting you in your gardening by planting some more in other spots. From our page on Morella cerifera (wax myrtle) here are the propagation instructions for that plant:

Propagation

Description: Sow seed outdoors in the fall or stratify. Wax myrtle will root from softwood or semi-hardwood cuttings taken in summer. Root cuttings, 2-3 in. long., can be made in early winter.
Seed Collection: Collect fruit in September or October. Leave waxy coating on during storage, but remove it prior to sowing or stratification. Remove in a solution of 1 t. lye to 1 gal. water. Store seeds in sealed, refrigerated containers.
Seed Treatment: Stratify in moist peat 60-90 days at 34-41 degrees. Many seeds have a waxy coat that prevents water uptake and stratification. Soaking seeds in hot water or rubbing them vigorously against a rough surface will helf remove the wax.
Commercially Avail: yes

According to this USDA Plant Profile, wax myrtle is native in and around Tuscaloosa County, so that raises the possibility that your new sprouts have been planted by seeds. What you are seeing could also be adventitious sprouts from the roots of your existing plants, but it doesn't seem very reasonable that the roots from the shrubs in your front yard would have extended to your  back yard. 

To help you in determining if what you have in the back yard is really wax myrtle, go to this Auburn University website Trees of Alabama and the Southeast. You might also want to read this previous answer about wax myrtle and male and female plants. Both male and female bloom, but only the female has berries, which many species of birds love. In order for the female to have berries, there must be a male of the species in the neighborhood that blooms at the same time. It would be interesting to see, if your small plants are, indeed, wax myrtle seedlings, if they are male or female. 

From our Native Plant Image Gallery

Morella cerifera

Morella cerifera

Morella cerifera

Morella cerifera