Yes they can! Clarinetists are often required to play quarter-tones in contemporary music and I've developed a fingering system to make this possible. My Dad was a composer who was very active in the New York Microtonal Society--among other things--and I've adapted and codified his fingering and notational system for a 24-tone scale on the clarinet. It does not require additional keys on the instrument--your regular clarinet can do it all! Watch the video and see and hear the possibilities. If you're interested in learning the fingerings or purchasing my quarter-tone fingering chart contact me through my website and I'll be happy to help.
In addition to contemporary music, quarter-tones are often used in different types of ethnic music, notably Arabic and other Middle Eastern styles. However, I've found that those usually sound more stylistically appropriate if they're achieved by adjustments from the embouchure and throat rather than fingerings on the clarinet. Having said that, though, I've written and recorded a Klezmer Freylekh -- a typical klezmer dance tune -- that uses quarter tones. You can listen to it here, recorded by my klezmer band, Big Galut(e). So, yes, quarter-tones ARE fun.