(Someone seems to have made a couple of mistakes in edits to the question. The curl of E appears twice.)
In a vacuum, and in a system of units where c=1, the two Maxwell's equations you're talking about look like this:
$$\operatorname{curl}\textbf{E} = -\partial\textbf{B}/\partial t$$
$$\operatorname{curl}\textbf{B} = \partial\textbf{E}/\partial t$$
There are really two questions:
(1) Should the two signs be opposite, or should they be the same?
(2) If they're opposite, which should be which?
The answer to 2 is that this is just a matter of convention. Instead of the thing we call the magnetic field $\textbf{B}$, humans could have chosen to define something, call it $\textbf{C}$, where $\textbf{C}=-\textbf{B}$. Under this substitution, the signs in the two Maxwell's equations both flip.
There may be more than one physical way to justify #1. One way is to observe that the combination of these two signs allows us to have solutions to Maxwell's equations that are oscillating waves, whereas if the signs were the same, we would have solutions that were exponentials. That is, the fact that the product of the two signs is $-1$ is what makes it a negative feedback system rather than a positive feedback system.
The version of Lenz's law that you cite is not a great way to think about Maxwell's equations. Induction doesn't really have anything to do with currents or circuits. But note that the formulation of Lenz's law as "resisting the change" is invariant under the change of variables from B to C, since it makes no explicit or implicit reference to any handedness. The "resisting the change" thing only depends on the product of the signs and actually requires the product to be -1.