Can you ever exert more downwards force than your weight?
The force you can exert is your mass times your acceleration. By the equivalence principle, just standing still is equivalent to accelerating at 9.8 m/s2, which is where the force of your weight comes from when you just stand still. But it is easy to accelerate more - like when you jump.
The force is only limited by your ability to push yourself off (transfer force to) the willow shoot. Imagine that you lie down next to the shoot, holding it in both hands. If you now pulled yourself up rapidly (the way some circus acrobats can pull themselves up a rope while appearing to hang horizontally) then you apply all your weight to the willow - and if you are strong enough to accelerate yourself while doing this, you could apply a force greater than your weight.
However, as you probably realize, there are other far more effective means to drive a stick into the ground. The key is to convert momentum into force - the equation is
$$m\Delta v = F \Delta t$$
This equation tells us that the change in momentum ($\Delta(mv) = m\Delta v$) is determined by the integral of force and time ($\int F\cdot dt=F\Delta t$ if F is constant). This is a direct consequence of the equation $F=ma$, which you can integrate with respect to time to get $\int F\cdot dt = \int m\ a\ dt = m \Delta v$.
When you use a hammer etc, you give it momentum during a long swing (small F, large t); but it slows down and comes to a stop in a very short interval, meaning that for that short time the force is much greater. A post driver is the tool people use to try to replicate this on the scale of large sticks being driven into the ground (hard to hammer the top of a tall thin stick). It may not be possible to use in your particular situation - but in general, it will allow you to apply a force much greater than your weight (bot for a shorter time). This is also the principle behind pile drivers etc . All these methods require the object to be driven to be strong enough to support the force you use to drive them into the ground...
Use a lever. For an application like driving branches in the great outdoors, you would need to come prepared, or locate the site next to something heavy. Anchor one end of the lever under something heavy. Attach something to the branch to press against, and put the branch between you, at one end, and the fulcrum at the other. The lever acts as a force multiplier, so you can exert more downward force than your own weight.
If you think about Newton's third law and standing still vs jumping. When you stand still the ground exerts a reaction force on you which is equal and opposite to your weight by Newton's third law.
If you jump upwards at the point where you begin to drive upwards you are applying a greater force on the ground than the standing still case, this difference is the reason why you accelerate upwards.
If you could somehow stand on the poles, squat down and jump explosively off them you would drive them into the ground with a greater force than your body weight. (If you could land on it again that would be even better!).