The second set of Blast Box minigames in Blast Rush LS -- Gust Gauntlet and Ship Toss -- are unlockable by trying out the main gameplay modes. Unlocking these also put you on the path to unlocking the other bonuses in Blast Rush LS! We'll post about those soon...
Here's the main criteria. You can complete these in any order:
Play both LS Mode game styles at least once. This means:
The main Standard and Endurance variants, not 2-Minute
In Standard: finish or get Game Over
In Endurance: Simply complete a run (get Game Over)
Play any Classic game style once. This means:
Finish or get Game Over in Standard, Endurance, or their 2-Minute variants
Play both Tier 1 Blast Box games once. This means:
You guessed it: finish or get Game Over in Time Attack and Flash Rush
Return to the main menu and you should see the notification!