See Wurth's post above ... should be next week.
Is the device being mounted by El Capitan (can you see it in Finder)?
Sorry about this. We have a fix for the problems and were trying to get it out before 10.11 was released, but obviously missed the date. 4.5.2 will be released next week and will support El Capitan.
You're going to want to contact Garmin Support then as this is either a device issue or an OS X issue. The device has to be mounted by El Capitan in order for BaseCamp to recognize it.
Will the update fix the spinning freeze we had with Yosemite?
I'm not familiar with the 276c, but it may be connecting in Garmin Mode instead of mass storage. That means it should show up in BaseCamp even when it doesn't show up in Finder. We're delaying the release of 4.5.2, so I can look into these Garmin Mode device connection issues on El Capitan.
I'm not familiar with the 276c, but it may be connecting in Garmin Mode instead of mass storage. That means it should show up in BaseCamp even when it doesn't show up in Finder. We're delaying the release of 4.5.2, so I can look into these Garmin Mode device connection issues on El Capitan.