Merriam Webster states that shivereens is the synonym of smithereens but there's no mention of the origin of the word.
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:
On the same page, Heslop also has an entry for shiverin boot:
So at least one possible interpretation of shivereens is that it derives from shiverings in the sense of "small shivers [of stone]."
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’)