As for the saying. What I think they mean by it is this:
Beer before liquor-
In other words your starting with a low alcohol content, moving on to a larger, in some cases much larger alcohol content depending on how its mixed. If some people dont know their limits that could cause disaster because you can get drunk faster than you think once you hit the hard liquor because of the fact that as you are drinking you are increasing the alcohol content as you go.
Liquor before beer-
In this case here your body is used to the hard liquor from the beginning. Moving on to beer is something like taking a glass of water after your done drinking. Your body is used to the high alcohol content from the beginning, moving on to a lower percentage as you go on could cause you to maintain your alcohol intake.

You say “drinking beer then liquor makes you drink a larger quantity of alcohol in the long run”

…not necessarily. For me, if i drink beer first there’s a chance i won’t even get to the liquor because beer is more filling.

The rhyme is saying that mixing alcohol in one order is less likely to get you sick than mixing it in another order and that’s simply not true. It doesn’t matter how much of each you’re drinking – that’s not the point of the saying. The assumption should be that the person drinking has no limit at all and could drink forever if they wanted (and who wouldn’t want that?). If the rhyme were true then that person would still get sick by drinking beer before liquor.

Because that person wouldn’t be sick just because they drank beer before liquor, the statement is false and is ultimately a myth. And, (before) now, a debunked myth.