Why Shyer Not Shier?

What Merriam-Webster is saying by including the comparatives shyer, shyest, shier and shiest is simply that these are words; it's not implying that more shy and most shy are ungrammatical.

Which is correct shyer or Shier?

They're all correct; both spellings can be used for any such word. However, in fact shyer/shyest is much more common than shier/shiest, but conversely driest is much more common than dryest. (The noun drier meaning a machine for drying may have different numbers.)

Is there such a word as Shier?

"shyer" or "shier"? Both versions are acceptable in today's standard English. In the 2002 CGEL page 1581: Monosyllabic dry and shy are optionally exceptions to the y-replacement rule, allowing either y or i before the suffix: dry ~ dryer/drier ~ dryest/driest and shy ~ shyer/shier ~ shyest/shiest.