Are egg rolls, common in Asian restaurants, such as Chinese, Thai, or Vietnamese (at least in America), supposed to be eaten with chopsticks or hands?
It's a matter of personal preference when referring to restaurants in the United States. Some egg rolls are too large to be easily handled with chopsticks, and they can be eaten using your hands; smaller ones and similar preparations such as spring rolls can too, but you may find it preferable to use chopsticks. When choosing your own egg roll from a common platter or service station, it's generally considered polite to use a provided utensil.
Note again that this applies only to Asian-American cuisine and American cultural norms. Etiquette may differ in other countries; when in doubt, watch what those around you do and follow their lead.