What does your game say at the upper-right corner of the title screen? It should have an "a" at the end of the version number. If it doesn't, you don't have Bunny Day activated.
I believe the game must be loaded with an internet connection to download this data (it literally just unlocks Bunny Day - no content to download, per se). Once it's downloaded, you're safe to play offline for the remainder of Bunny Day. You still have enough time to get the eggs to craft everything.
I've been getting cherry blossom DIY recipes from the odd balloon. They don't seem to be very common.