I am trying to pack bytes into as few unicode characters as possible. Consider the Twitter 140 character limit, which allows byte-heavy unicode characters.
I don't think it's as straightforward as this. Consider that there are lots of unassigned Unicode code points, different kinds of whitespace, nonprintables, and so on, all of which may or may not be removed/replaced/folded depending on where you're posting this text. This means you'd have to look at the character properties, select only those characters that are likely to work correctly, and make a mapping.
I believe I can squeeze 3 8-bit ascii characters into a single 20-bit unicode character
Well, to nitpick, ASCII is 7 bits, and 3*7 is 21, while Unicode code points range from 0 to 0x10FFFF (1114111), which won't fit 2^21==2097152. If you exclude ASCII control characers, for example, you'd also be excluding Tab, CR, and LF. afoken went into more details.