Forces while squeezing a toothpaste tube When I squeeze a tube of toothpaste, I am working with 2 squeeze forces toward tube. What causes the toothpaste to go out of the tube? (instead of remaining stationary)

 A: $$dL=pdV$$
You are changing the volume and thus imparting work on the paste. It's incompressible and must flow out. 
A: From a fluid dynamics perspective, your applied force induces motion via the Navier-Stokes equations:
$$
\rho\left(\frac{\partial}{\partial t}+\mathbf v\cdot\nabla\right)\mathbf v\propto\mathbf f_{body}
$$
where $\rho$ is the density of the toothpaste, $\mathbf v$ the velocity of the toothpaste, and $\mathbf f_{body}$ your applied force. Since the density is non-zero, the only way for the equation to balance is if $\mathbf v\neq 0$; it will then flow out the only orifice it has.
Alternatively, you can view this as your squeezing is shrinking the volume of the container, which must displace the material filling that volume.
A: Easiest said: a force from the side causes an increase in pressure. In simple (Newtonian) fluids and gases, pressure is a scalar field. You may have a different pressure at each point in the fluid, but there's no direction associated with that pressure. It's equal in every direction, including up and down. Now on the bottom there's a wall resisting the downward pressure, but not on the upper side.
The analysis is somewhat complicated analytically because toothpaste famously is not Newtonian. It appears quite solid when not under pressure. That's why it doesn't drip of your toothbrush. But when you apply pressure, as you do here, it becomes more liquid and flows more easy. 
A: Your free body diagram neglects to include the force from the back of the tube on the toothpaste as well. This force is not negligible and is comparable with the applied force from your hands. It is the imbalance between this rear force and the integrated pressure force on the front of the toothpaste that results in the motion.
A: in simply way,  when you applied force from both direction the pressure inside the tude increases. As we know that air moves from higher pressure to lower pressure. So the inside air moves outward carrying paste too.
