ChristiaanJ is completely right - the faster you start the 'zero G parabola', the longer it lasts. But to get maximum time from it, you would have to maximise the vertical component, much as Mark 1 and chornedsnorkak have implied.
It would appear that your NASA engineer was wrong, or you misunderstood him, as initial vertical speed is EVERYTHING! Shoot a projectile upwards at high speed, and I promise you that the faster you shoot it, the longer it remains at zero G. In the extreme case, send a projectile around the earth at the sufficient speed, and it will remain in zero G orbit indefinitely.
The shape of the wing and the CG position as brought in by Slasher, including random figures of M.98, 30 degrees and 25 degrees are complete red herrings, which inevitably lead to an incorrect conclusion. Even a stone or an Aeronautics text book achieve zero G every time you throw them (provided you never open the book!), regardless of their shape or C of G position.
If anyone were to be serious about CJ's proposal, many other limitiations would need to be considered, eg wind loading, max altitude limitations etc. Of these, not least would be the fact that there are no airworthy Concordes available, and it is doubtless too small to be able to do anything useful with whilst weightless anyway!