Meissner Effect and Lorentz Force Paradox?

A great question!

The answer is that there is a still a force of ${bf I}times {bf B}_{rm external}$ per unit length of the wire. This is because the ${bf B}$ field does penetrate some distance (naturally this is called the penetration depth) into the supercondcuting wire, and, for reasons similar to the Meissner effect itself, this near-surface penetration depth region is also where the current carried by the wire flows. That the location of the current and strength of the penetrating ${bf B}$ field conspire to give exactly the same answer for the force as if there were no Mesissner effect is not exactly obvious. It is, however, a magnetic analogue of the statement that if you put a charge $Q$ on a conducting body and immerse the body in a uniform electric field ${bf E}_{rm external}$ then the force on the body is still exactly $Q {bf E}_{rm external}$ despite the fact that there is no ${bf E}$ field inside the conducting body.