Another better way of figuring this out rather than to try use Lenz's Law is to imagine a test positive charge sitting on the metal disk and apply the Lorentz force law (F=qvB or F = IBL).
The test positive charge is moving to the left when the disk rotates into the magnetic field (so I, the current, points to the left).
The magnetic field points into the page.
Line up your hand accordingly, and you'll see that the force points down the page (this explains why the eddy current goes down for the part that is in the field).
Of course, since charge is conserved and shouldn't pile up anywhere it has to loop around in the part that is outside the magnetic field.