To someone who is not so experienced in advanced physics, how do you explain or teach them why something as complex as color confinement prevents us from being able to see or prove that there are any type of quarks existing in the quantum level and why not even with the strongest microscopes are they visible.
I agree with John Rennie, that you should read the article about confinement, but I will still attempt to answer what you may want to explain/teach to give your students a reasonable overview.
A good point to start would be the phenomenology. Don't make particle physics a black box. Show what we actually "see" in accelerator experiments! For the purposes of this problem we are seeing heavy particles called baryons (the most common ones are the proton and the neutron) and mesons.
Reactions with these particles happen to conserve the so called Baryon number, which is $B=1$ for baryons, $B=-1$ for anti-baryons and $B=0$ for mesons. $B$ is a conserved quantity, i.e. a good quantum number which behaves similar to an electric charges.
Proceed by explaining that physicists also noticed that one can identify additional quantum numbers like the Isospin $I_3$, which behaves similar to an angular momentum quantum number, Charm $C$, the Strangeness $S$, Topness $T$ and Bottomness $B'$. These are called "flavour quantum numbers". Flavour quantum numbers happen to be preserved in processes that happen due to electromagnetic and strong interactions, but they are not preserved by the weak interaction.
Now you can show that with these quantum numbers one can then sort all of the observed hadrons and mesons into multiple diagrams similar to the periodic table or the table of isotopes. If you can afford the time, show the similarity. It follows from the underlying symmetries of this order that hadrons can be understood as compound particles made up of three sub-units, while mesons are made up of two. The subunits were called quarks, the particles carrying the strong force between them were called gluons. Overlay the quark content of each particle with your original hadron/meson diagrams. Let students experience how the combinations of two/three quarks add up to a neat sub-nuclear puzzle. This should give students some confidence, that the quark idea is maybe not so bad, after all!
Once you have established that confidence you can say that curiously not a single quark has been seen in any one of these experiments (they would give themselves away by their fractional charge!).
And this is the point where the confinement hypothesis comes into play. By defining a suitable interaction between the quarks one can find a field theory which predicts, that any attempt to create a free quark will automatically produce new quarks in the process, i.e. the resulting particle will always be a meson or a hadron, but never a quark. This is where the pictures of snapping rubber bands may come handy.
One can, of course, leave a few of these steps out, and show the snapping rubber bands right away, but it may be better to give students at least a glimpse of the full picture, rather than just a random fact about confinement.