Can a Rutan Quickie be made into a legal Part 103 Ultralight?
I doubt the Quickie would fly well enough at very slow speeds to meet the rest of the Part 103 requirements -- specifically, the limit on maximum level flight speed.  Limiting fuel capacity is easy, and the 18 HP original engine worked well enough on the Q1 -- but it was a lot faster than 55 kt (close to double, as I recall).  From comments (thanks, Gerry) I understand the Q1 has a stall speed of 42 kt -- fairly reasonable for a light aircraft, but well above Part 103 maximum of 24 kt.

You might be able to slow down enough by installing a flatter-pitch propeller, but that would have other effects (overspeeding the engine, for instance) that would need to be addressed, and then you'd be trying to fly with almost no speed range -- 42 to 55 kts isn't, IMO, a safe performance envelope (and yes, the Q1 would still go faster than 55 kt in a dive, but the level flight and climb performance are where the safety issues lie in this case).