Merriam Webster states that shivereens is the synonym of smithereens but there's no mention of the origin of the word.
2 Answers
According to the OED, it is simply an alternative form of ‘smithereens’, influenced by the noun ‘shiver’, which is in the same semantic area and phonetically quite close.
(‘Smithereens’ is of course a naturalised English plural form from Irish smidiríní, a diminutive of smiodar ‘scrap, fragment, tiny piece’)
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Shiver has come up before on ELU. I don't recall seeing/hearing shivereens before (and it basically flatlines against smithereens in Google Books), but apparently it's been bubbling around under the surface for an awful long time. – FumbleFingers Jul 26 '13 at 21:16
Oliver Heslop, A Glossary of Words Used in the County of Northumberland and on the Tyneside (1892), published for the English Dialect Society, includes the following entry:
SHIVER, a splinter, a flake of stone. Shiverins, Shivereens, very small fragments. "It's gyen aall te shivereens," equivalent to the colloquial "all to smithereens."
On the same page, Heslop also has an entry for shiverin boot:
SHIVERIN BOOT, a shivering fit.
So at least one possible interpretation of shivereens is that it derives from shiverings in the sense of "small shivers [of stone]."
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