Hi Cliff
The libraries under Bridge are intended for the Yun and the Tre only.
This page on the Temboo website talks about writing programs for other Arduino boards with the ethernet shield - it is for the example you mentioned with Google spreadsheets.
It sounds like you configure stuff on Temboo and then it generates a skeleton Arduino program to download. If you can do that and post the program, it should be possible to modify it to work with the ESP8266 wifi stack rather than the Arduino ethernet shield.
Regards
Ray