I have used MapSource on Ubuntu with Wine. I think it mostly worked, at
least for viewing maps / waypoints / tracks etc. I'm not sure if
uploading/downloading to the Garmin worked.
I've never tried EasyGPS.
You could check the Wine AppDB, though most of the reports are for old
versions of MapSource and Wine.
Depends on what you want to do, but it may be better to use some native
Linux software instead.
eg for an alternative to MapSource, try QLandKarte GT. It can load maps
in Garmin format, plus it can display waypoints / tracks etc.
For transferring data to and from the Garmin, you can use GPSBabel. It
is also very useful for converting between formats.
You could also try Viking or GpsPrune, these let you view tracks on top
of a variety of maps.
I think all of these are in the Ubuntu repositories, so you should be
able to install them from the Ubuntu Software Centre.