Want to connect non-bose speakers to a bose lifestyle system

this is possible, however not a very good idea.
Bose speakers are designed a little different than other brand speakers, the Equalizer built into the bose system will probably not sound amazing when you hook up your other speakers. Are you looking for more bass? cleaner highs?

if you want more bass, you could try doubling up your subs, if you want cleaner highs you could buy Bose Jewel Cube Speakers (purchased from a Bose Built-Invisible Dealer