You need to find FIVE eggs BEFORE you will get a package.
You should get the package immediately after you find the 5th egg.
You will know if you have a package because a little brown package will appear under your watering can.
It takes about 10 minutes or so between eggs. So it should take almost an hour of collecting eggs I think. (It did for me at least. It may be different for others.)
It also appears you can only get one package because I haven't seen any more eggs after I received my package. (I suppose you could get a second Egg Flower if someone gifts you theirs.)
And typing that I just had an idea of what the 'flower' might turn into. I'll be quiet and see if I'm right. Don't want to spoil it if I am.
EDIT: oh, just read the above post. You found more eggs? Awesome! I wonder if we can get a second Egg Flower then. Off to look some more! And I already found that one.
EDIT2: Just found the 6th egg. I wonder if there are more places to look. I shall scour (look everywhere) I possibly can!
EDIT3: Yes! Up to 7 eggs found. Will this get really annoying if I keep posting this? Just tell me to stop nicely and I will. But it appears there are at least 7 places to find an egg now. And you don't find the eggs in the same place. Like if you find one IN the Wild Garden, you won't find it there again.
EDIT4: 8th egg. I figured there would be an egg in this location and I'm right. Now to find out if there is one at all three or just one in general.
EDIT5: Okay up to 9 eggs found. There are at least NINE places to find eggs now. I'm keeping them documented too in a spreadsheet.
EDIT6: Found my 10th egg. Unfortunately at the time that I clicked on it I had 4 seedlings. So I don't know if I didn't get a package because of that or if you are only allowed to get one package. I'll go ask someone who should know.