Actually, Evertonians claim that Liverpool was formed as a breakaway from their club, although it was they (Everton) who moved out of Anfield (originally their home), thereby leaving it vacant for the newly-formed Liverpool FC.
I know it's from Wikipedia, but I think the following is reliable:
"Liverpool F.C. were founded by John Houlding in 1892 to play in his vacant Anfield stadium. For the previous seven years the stadium had been used by Everton F.C.. However, in 1891 Houlding, the leaseholder of Anfield, purchased the ground outright and proposed increasing the rent from £100 to £250 per year.[8] The Everton members objected, left Anfield and moved to Goodison Park.
With an empty ground and just three players remaining, Houlding decided to form his own football club and on 15 March 1892, Liverpool Football Club was formed. The original name was to be Everton F.C. and Athletic Grounds, Ltd., or Everton Athletic for short, but was changed to Liverpool F.C. when The Football Association refused to recognise the team as Everton. John McKenna was appointed director and signed thirteen Scottish professionals for the new club"
Anyway, I accept that that doesn't really matter for the purposes of this thread; your second paragraph is much more pertinent (and interesting).
It's beginning to look as though ROI fans have mixed, even contradictory, opinions on the issue of the Ireland team, pre-partition.