Our bodies' or our body's

"Our bodies" refers to a collection of bodies owned by a group of people including the speaker. If a group of people share only a single body, then "our body" would be correct. When speaking of an actual physical body with an immune system, it's extremely rare for it to be occupied by more than one person.

"Each of [our|your] bodies" similarly refers to that same collection of bodies. As such it is still plural; even though it is encouraging us to consider every member individually in sequence, the end result is that we are considering the entire collection.

However, if you're referring to "the human body" as a generic construct as opposed to a group of specific people and their physical incarnations, that is an acceptable time to use the collective possessive form "our" and still use the singular "body":

Now consider humanity. Our body is a wonderful, complex thing. Our body's immune system alone is a system of such vast complexity that supercomputers only now are beginning to model it successfully.

It could be either "our bodies' immune systems" (the plural possessive) or "your body's immune system" (the singular possessive).

Note that if the plural form is used, then it would have to be "systems" to agree in count.

"Our body's" simply does not work here unless one of the parties is non-human and can occupy the other's body.

In cases like this, it's worth considering what happens with a noun that has a different (phonetic) form in the plural vs possessive:

• this man's immune system
• ?these men's immune system
• these men's immune systems

So I would go for either: "our body's immune system" or "our bodies' immune systems". As discussed in another post, English has a tendency to prefer a "distributed plural" in cases like this (but other languages don't necessarily). So just as you'd say "The students got their pencils out", even though each student had a single pencil, it would be common to say "our bodies' immune systems" with everything in the plural.