The Lady Bird Johnson Wildflower Center is committed to the use, care and propagation of plants native not only to North America but to the area in which they are being grown. The only cannas native to Texas are Canna glauca (maraca amarilla) and Canna flaccida (bandanna of the Everglades). Canna indica is native to Central and South America and most of the other cannas are grouped under the name Canna x generalis, so extensively hybridized that there is no telling what the parents of a particular plant were or what their general characteristics will be. Both being hybridized and being non-native put them out of our range of expertise.
Since Canna x generalis (Floridata), as well as other named cannas, spread by rhizomes underground, mesh to keep the leaves from poking through is not going to make much difference. In southeast Texas, those rhizomes will continue in the soil over winter, and sprout new plants next year. We have heard complaints about cannas "taking over," but found no indication in our research that they are considered invasive. If your neighbor dislikes the cannas, she can dig the rhizomes out on her side as they come up. If you care about your neighbor's opinion, you can move the canna rhizomes to someplace on your property not so near theirs. This can be done over the winter.