Standard Cryptic Eagles still end a run on an unprotected hit from the front. Boss eagles are the bigger, redder birds. They spawn on their own cadence later in a run, fly faster, and show a three-pip health row above the body.
A blue shield token is how you actually fight them. Each shielded bump knocks the boss back and removes one pip. Three hits and the boss is gone. Those kills feed career XP and weekly Boss Hunter challenges.
What not to assume
A boss scrape is not always an instant Game Over the way a normal eagle is, but that is not a plan. Dodge still slips you through a body for a moment. Falling off the course still ends the run. If you see a boss and you have a blue token in range, take it.
Bosses do not change how platforms generate. They sit on top of the same gaps, stacked decks, and moving ledges. Learn the course first, then pick the fights.
Where they show up
Bosses appear later in a run on their own timer, not because you entered a special mode. The three-pip row above the body is the health you are playing for. A shielded bump knocks the bird back so you can line up the next hit without eating a gap.
Those kills feed career XP (30 XP per boss on the Next Level legend) and weekly Boss Hunter jobs. If you are not playing for the fight, dodge through and keep running. The mile counter is still the board.