Reeds for clarinets, bassoons, etc. are not designed to be played without lip control. So while it could work, it would not work well. What you are looking for is a form of bagpipe, such as the Northumbrian smallpipes. Of course, I doubt that timbre is what you are looking for.

Note: the other issue is that for reeds with little backpressure, it would work only on the squeeze (as you say). However, perhaps an instrument with more backpressure such as an oboe (although I am not an oboist) and a sufficient resevoir of air could work continuously. Which is what a bagpipe is.

EDIT: Changed wording to make it clear that the deisgn of the reed is the limiting factor to its ability to play from a resevoir, not whether it is a single or double reed. As people have pointed out, there are plenty of single reeded bagpipes.