Yes, eggs are naturally gluten-free.
However, eggs are often at a high risk for cross-contact due to the ways they are prepared.
The way eggs are prepared (such as scrambled or fried) often puts them at a high risk for cross-contact, especially in restaurants. Eggs are often cooked on large, commercial griddles or in a pan that may have been used to also prepare gluten-contains foods..
Additionally, spatulas, forks, or other utensils that have touched gluten also greatly increase the possibility of cross-contact, and would be enough to cause a person with celiac disease to have a reaction. Finally, if a restaurant is also making pancakes, muffins, or anything with wheat flour, barley, or rye, there is an additional risk for cross-contact .
Please be aware of the risks of eating eggs at restaurants that also prepare gluten-containing foods, ask your server specific questions about how they will be prepared, or ask if you can watch the food being cooked. Some restaurants have even been known to put pancake batter into scrambled eggs—so always ask questions!
If eating at home and taking the normal precautions for cross-contact, there should be little to no risk.