I am looking for a limerick or childs saying from the 1940's. It goes
something like this....Ona miori me ick me ann phyllison phollison
niquish john queasy quavy english navy stinklum stanklum be bi bo buck
you're out.

Dear goober,

The verses you have in mind are a playground rhyme children used for
"counting out" games. And this special rhyme dates back, in fact, much
earlier than to the 1940s.

Since children's rhymes have been passed down in oral tradition only
for a long time, there are of course many variations, but with
cognizable connection.

The oldest printed source I was able to locate mentioning a variant of
this rhyme is from 1866, and the author was Mark Twain, who wrote in
his newspaper article "Spiritual Insanity" the following:

"(...) Mr. McCoppin (counting on his fingers): 'One ery - o'ery -
ickery - Ann; fillisy, fallallacy, Nicholas John; queevy, quavy,
English navy - stinklum, stanklum, Buck. Alas, my poor, poor country.'
(...)"

Source:
Twainquotes.com: Mark Twain - Spiritual Insanity, by Barbara Schmidt,
2002
http://www.twainquotes.com/18660200t.html

This rhyme can be identified easily as a slightly altered version of
the one you remember. There are, however, more variations, like this
expanded one:

One riori - ickery ann - philiston pholiston - nicholas john - eavy
quavy - english navy - stinkum stankum - bonk - ini mini - miney mo -
catch a tigger - by the toe - if it hollers let it go - my mother said
to - pick the very best one - and it is why oh you

Source:
Bigego.com: Big Whoop, by Jim Infantino, 1997
http://www.bigego.com/lyrics/bigwhoop.html

A Boy Scouts website lists two versions of the rhyme, together with a
description of how this and other counting out rhymes were used in
children's games of long-gone days:

One-ry, or-ry, ickery, Ann! 
Fillison, folliso, Nicholas, John. 
Queevy, quavy, English Navy, 
Stinckelum, stanklum, buck!

- or -

One-ery, two-ery, hickory, han, 
Fillison, Follison, Nicholas, John. 
Queevy, quavy, Virgin Mary, 
Stingelum, stangelum, berry buck!

Source:
The Inquiry Net: Counting Out, by Dan Beard, 2002
http://www.inquiry.net/outdoor/summer/counting_out/

Or this one, recorded in 1885 by H. Cartington Bolton:

One-ery, two-ery, ickery Ann,
Fillin, falling, Nicholas, John
Que-ever, quaver, English knaver
Stinhilum, Stanhilum, Jerico, buck

Source:
Celtic Folklore: Sacrifice
http://www.belinus.co.uk/folklore/Baring/Bookfolklore05.htm

As it is mostly the case with texts of oral tradition, there is not a
defite "right" or "original" version. There is not even evidence on
when this rhyme developed. But since Mark Twain used it for his
article in 1866, one can assume that these words have by then already
been old and well-known to the public, so the readers understood the
rhyme without any explanation. The actual roots of these words might
be much, much older. There is some evidence that the origin of this
children's rhyme might be very old magic spells, as shown on this
website:

The Internet Sacred Text Archive: Chapter XIV - A Gypsy Magic Spell
http://www.sacred-texts.com/pag/gsft/gsft16.htm

Search terms used:
stinklum stanklum
://www.google.de/search?hl=de&lr=&ie=UTF-8&oe=UTF-8&newwindow=1&q=stinklum+stanklum&sa=N&tab=gw
Eavy Quavy English Navy:
://www.google.de/search?q=Eavy+Quavy+English+Navy&ie=UTF-8&oe=UTF-8&hl=de&btnG=Google-Suche&meta=
"nicholas john" "english navy":
://www.google.de/search?hl=de&ie=UTF-8&oe=UTF-8&newwindow=1&q=%22nicholas+john%22+%22english+navy%22&btnG=Google-Suche&meta=
"Counting-Out Rhymes" bolton:
://www.google.de/search?q=%22Counting-Out+Rhymes%22+bolton&ie=UTF-8&oe=UTF-8&hl=de&meta=

I hope this was informative for you.
Best regards,
Scriptor

Thank you so very much. Our family has been saying this for so long
and we finally know not only the correct spelling but the origin as
well. This was worth the effort and money....thank you again!