Depends on how you define 'evergreen' and what type of hellebore you are dealing with. 'Evergreen' as it pertains to perennials or other non-woody plants is a bit different from the evergreen of shrubs and conifers. Virtually all 'evergreen' perennials benefit from being cut back or having old foliage removed on a regular basis, generally once a year. The oriental hybrids, Helleborus x hybridus, hold their foliage pretty well throughout the year but by late winter this foliage is pretty tattered and weather beaten. And it often carries overwintered disease organisms. Most growers remove it just as the flower buds - carried on separate stems - begin to emerge from the ground in late winter. If not removed, it will die off naturally that season, but it improves the plant's appearance to remove it before that happens.

Caulescent hellebores or those that bloom on leafed stems, like many of the species and lots of the newer hybrids, seem to hold their foliage better in winter. But these old flower stems also need to be cut back each year (to the base of the plant) after the bloom period.

In both acaulescent and caulescent types, the old foliage is rapidly replaced by new growth.

Any plant can be grown in containers if given the proper care, but most hellebores and xhybridus in particular would not be my first choice for container plants. In Texas, I'd be concerned about the placement of the containers during the hot months and making sure they received adequate water.