Question

Why BeCl2 is covalent and other MgCl2, CaCl2 are ionic

Answer 1

beryllium is smaller atom than magnesium and shows less tendency to lose electrons.
so therefore
becl2 is covalent and mgcl2 and cacl2 are ionic

Answer 2

A chemical bond between atoms of different elements in a compound can be mostly covalent, some covalent and some ionic, or mostly ionic in nature.  That depends on the extent of electron sharing that happens between the two atoms.

Covalency means sharing of electrons,  Ionic means giving away the electrons or holding the electrons.

Ionic nature increases with increase in the size of cation or metal.  As the electron is loosely bound to the nucleus when the radius is large.   Covalency is also affected by with the charge present on the cation. Large charge present on cation can attract the electron. 

Beryllium , Magnesium and Calcium are in the group 2 and alkaline earth metals.  Size of Beryllium is less than that of Magnesium. Magnesium atom is smaller than that of Calcium.  They all have same charge on the atom/ion.

So Be - Cl bond is most covalent, Mg - Cl bond is less covalent and some ionic, and Ca - Cl bond is most covalent among the three.