Is it true that Quadratic residue was published and discovered before Legendre symbol and Euler's Criteria? So Is it true that Quadratic residue was published and discovered before Legendre symbol and Euler's Criteria?
Quadratic residue came in 1801 by Gauss(1). 
can you put these concepts in chronological order and by whom these concepts was first introduced or developed in paper form?
I'd say yes. My reason being that the Legendre symbol simply defines if a number $\mod p$ is a quadratic residue to $p$ or not. There is no way to define a number being a quadratic residue $\mod p$ if you don't even know what a quadratic residue is.

Regarding the Euler criterion, that's a little bit more tricky. That too would've come after the discovering quadratic residues, but I'm not sure if that was before or after the Legendre symbol
I'd say yes. My reason being that the Legendre symbol simply defines if a number $\mod p$ is a quadratic residue to $p$ or not. There is no way to define a number being a quadratic residue $\mod p$ if you don't even know what a quadratic residue is.

Regarding the Euler criterion, that's a little bit more tricky. That too would've come after the discovering quadratic residues, but I'm not sure if that was before or after the Legendre symbol