I have both Chromecast and Global Go on my iPod Touch; not exactly a fan of watching an hour-long TV show on a very small screen. So, I clicked on the "Send Feedback" button on the app and asked if they were thinking of supporting it in the future. Here's the response, FYI.
We appreciate your interest in using Chromecast with our app, currently we do not support Chromecast on our Shaw Go apps. The present design of Chromecast has a feature to buffer the stream and save a portion of the video onto the internal memory of your device. This presently violates our license and DRM as the app cannot be downloaded onto devices, it is available to stream only. In future we may support such a feature, if Chromecast has any updates to disable such a feature we may also be able to support it.
So, it looks like a licencing issue, primarily. I think he meant, instead of saying, "the app cannot be downloaded onto devices" (erm, I downloaded/installed it onto my iPod, no?), something about data not being allowed to be downloaded onto a device. Or something. I dunno.
Anyway, that's your answer.
Oddly, as someone else on these forums has mentioned, the CTV Go app works well with Chromecast. Perhaps they were able to wrangle a different deal.