Seems like NP cannot equal coNP by the definition of NP
This kind of argument presupposes that the only way to (for example) prove that a SUBSET-SUM instance is not solvable is by going over all subsets and checking that their sum doesn't match. This isn't true &mdash; a meet-in-the-middle algorithm works in time $O(\sqrt{2}^n)$ rather than $O(2^n)$, where $n$ is the number of sets.

Furthermore, consider the analog of this question for polynomial space rather than time. In this case it does hold that NPSPACE=coNPSPACE, for the simple reason that PSPACE=NPSPACE.