Is everywhere a proper noun

No, the noun 'everywhere' is a common noun, a general Word for 'all places'.

A proper noun is the name or title of a specific person, place, or thing; for example, "Everywhere' by Fleetwood Mac or SportEverywhere (travel consultants) in Verona, Italy.

It should be noted that only two of the eight dictionaries consulted designated the Word 'everywhere' as a noun.
All of the dictionaries consulted agreed that 'everywhere' is an adverb.
One of the dictionaries consulted designated the Word 'everywhere' as an adverb, a pronoun, and a conjunction.