Having been in Germany for a couple of years defending you slackers [and civilization in general] from the “Evil Empire”, IMHO the origin of “Ollie, Ollie oxen free” is simply an Americanization of the German, “Alle, Alle auch sind Frei.” Literallly translated, this is, “All, All also are free.” Ollie Ollie probably came about when immigrant German children and American childred played together in cities like N.Y. in the early part of the 20th century.
The word is “s’enfuir” (unless it’s old French or something), no space between “en” and “fuir”, meaning to flee or run away
However, wouldn’t running away be in the wrong direction for Hide and Seek?
The phrase you are discussing is not normally found among English speaking children on this side of the Atlantic. It is used mainly by children influenced by an American background. (This contrasts with rhymes like “Ring a ring a roses” which are part of a shared cultural background.)
This lack of background supports the belief that it is a German phrase - that it was imported by children as uncomprehended sounds into the American language. That is why there are so many variations of the phrase, since it was originally heard as a series of nonsense syllable.