This all boils down to personal preference. Actually 1/4" plywood panels
would be fine. While you may be able to push on the panel and deflect it,
it adds no strength to the panel as a whole. It simply fills in the hole
between the rails and stiles.

Thanks.

If however you are going to attach any thing on the in side side to the
panels you want at least thick enough to handle screws if you will be using
screws. Personally I would choose plywood over solid wood for stability,
and ease of construction.

I totally agree that plywood would be more stable and easier, however I feel
that the grain of plywood sometimes does not look natural. This is the
primary reason that I want to go with solid stock.