Ive got whiskey, apple, and cherry chips. Leaning toward the whiskey chips.
Thinking 250* for 1.5-2 hrs with some beef stock in the pan.
Whiskey chips are from oak, which is a bit heavier smoke, so that should be a good pairing with your smoke subjects. You could mix it up with cheery and/or apple as well...cherry being a bit heavy, apple being lighter smoke. Dunno what Cavender burgers are...something I should try smoking? But of course anything I haven't yet smoked would be something I should try, right? LOL!!!
Beef stock should add a bit to the overall flavors, and, can be used for the finished burgers when the smoke clears, with a little creativity, adding that one-two punch on plate.
I don't think your eggs will take quite that long @ 250* chamber temp to reach minimum safe internal temps (165* for ground meats), but if you have a therma-pen, it'll give quick readings on smaller items. In any case, a probe check on a few sample eggs should give you a good idea where they're at.
Let us know how it turns out for you, and show us a little q-view, if possible...always nice to see what everyone's up to, when I can. Have a great smoke!!!