Let's say for example, Jack is pushing on a box with force $F$, and John is pushing from the opposite side with force $F$ as well. We say that there is no resultant force, and the box remains stable in its place, right? However, when a car is moving at a constant speed, the resistance is equal to the force that drives the car, so why is the car still moving? Isn't it supposed to be stable if there is no resultant force?
You are using the word "resultant" in place of a more precise mathematical term - "net", or "sum". Add up all the forces on an object, and if they equal zero, the object experiences no acceleration. For your box, since the forces are in opposite directions, $$\sum F_i=F-F=0$$ For the car, the forces are actually quite complicated, but start with the fact that the car is "moving at a constant speed". That means it has no acceleration (assuming one-dimensional motion). No acceleration means no net force, so the car apparently has no net force on it. All the force "pushing it forward" is being counteracted by the force "pushing it backwards" (air friction, ball bearings, etc etc etc).
When you say the word "stable", you mean "no acceleration", or "no changing velocity", not "no force".
As Galileo said, movement is nothing. There is no absolute rest of movement, only things moving with respect to each others. A Force is something changing the "state of movement" in a galilean reference frame. The earth surface is such a frame with an acceptable approximation in your car example. Therefore, when all forces (thrust from tyres coming from the engine, various frictions, weight, perpendicular reaction of the horizontal road, for example) sum to a nul vector and without torque, your car will keep the same "state of movement", i.e. velocity, with respect to the ground.
It is quite complicated to explain but the thing is that the tire of the car is performing CTRM( combined rotational and translational motion) and due that opposite pair of forces (frictional and that of car ), there is no relative slipping of tire on the ground. Movement of the car is due to rotational motion of tire.