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Hi WarrenSteph and welcome to MySwissAlps!
This page might answer your questions:
http://www.sbb.ch/en/station-services/car-bike/on-the-move-by-bike/take-your-bike-with-you.html
To look for what applies to specific services, do a timetable search for the trip from Zürich to Andermatt on your intended travel day. You will need to change trains several times, and different rules will apply to each train. Symbols in the comments column will indicate what applies to any given train (see attached screenshot for explanations of symbols).
You will need to look for connections that permit bicycles all the way from Zürich to Andermatt, and some of the trains may require compulsory reservations for bikes.
http://www.myswissalps.com/ti metable
Alpenrose