Last I knew Google Fiber used PON. With PON a provider would not allow a subscriber to use their own equipment.
The above is important because PON is essentialy one bidirectional fiber from a central office serving dozens of customers in the field. If you plug in with something on the same wavelength you could interrupt all the other customers on your PON. In PON since your fiber is not dedicated all the way back to the CO, they would have no way to know what device was trying to connect to their access node. They would need to configure your devices Mac address similar to how DOCSIS cable internet works.
If they ARE using dedicated fibers back to the CO then that would more likely be Active Ethernet, a slightly different technology. They would probably still tell you no because of how the CPE (the box at your house) needs to be remotely configured by the access node for shapers etc.
Is it possible that you have a power issue at your house that is causing the failures?