What is the recommended temperature for cooking eggs?
The FDA recommends 160°F (71°C) for dishes containing eggs. If you're scrambling eggs or making an omelette, as long as they're coagulated/solidified, they'll be safe.

The easiest way to confirm this is with sous vide eggs. For example, this Food Lab sous vide egg guide has 155°F, 160°F, and 165°F eggs:

  

which clearly show that the white won't be totally solidified until 160°F or even 165°F, even when cooked for an extended period of time.

(To be clear, those lower-temperature eggs are actually safe if cooked for sufficient time. The point here is just to show coagulation versus temperature, to demonstrate that when eggs cooking in a pan, you can use the solidity as a temperature indicator.)