Thanks to the hint by @vicatcu here, I was able to get my long dormant eggs online again. The key was finding a system that could accept http requests since the Arduino / Nanode doesn’t have the resources to make an https request.
Adafruit io does allow http requests and was relatively easy to set up. I created a free account (up to 10 feeds, 5 dashboards, 60 calls / minute, 30 day storage) and a feed for each sensor (Temperature, Humidity, CO, NO2, and Dust).
Next was modifying the firmware for the eggs. I forked the v1 repository in order to track changes in my own repo. First I decided to leave the remote egg alone for now and focus on the base. I played around with the EtherCard Library examples and eventually was getting successful gets and posts with the adafruit io api. I made a copy of the AQEBase sketch and went through commenting out parts until I got to the call to postSensorData() method. I made a postSensorDataToAOI() method that checked the sensor type and set up the feed accordingly. The code I ended up with is really nasty, but it is working. I hope to clean it up soon.
Glad to get this thing back online. If anyone has a v1 ‘kickstarter’ egg that they want to try and get back online, let me know.
Big thanks to Wicked Devices and especially @vicatcu for writing such nice arduino code. It was really easy to follow and find where to modify things.