There are a few ways to partially accomplish what you are after.
You could download gpx files of each segment you want to see and them upload them to your edge. It cnn display multiple gpx files at once. You will see them on the map but I don't know of a way to have it tell you one is coming. You just have to watch the map.
You could also just set a waypoint for the beginning and end of a segment. Then see those on the map.
Visiting St george/Hurricane? Stay at my vacation rental. Discounts for MTB's
I just came across this post. I actually created the Skidmap Segment map. The problem is that the old API would only let you get segments by popularity. We had a difficult time finding a good average between popular enough, but not to many overlapping in one place. And not matter where you go, segments popularity is almost always dominated by road segments.
Once Strava created the "Favorites" feature we could have easily created a custom basemap for the Garmin 800/810 that would allow users to overlay their favorite segments, as many as they want and it would be fast too, because I created it like a regular basemap. Then Strava decided to shut off their API. I even offered to created the Garmin/Strava basemaps for free for Strava, but no. I guess Strava doesn't want their customers to have features for their Garmin, even if free, or maybe they want to provide all the features. Not sure.
But I have all my favorite segments in a nice transparent basemap on my garmin, because downloaded them and made my own map. Just no point in coding up the automation, with Strava cutting off their API access and making my code worthless. Sorry I couldn't hook you up!