Can an equlateral triangle have vertices at integer lattice points?

No.  

Suppose 2 of the vertices are (a,b) and (c,d), where a,b,c,d are integers.
Then the 3rd vertex lies on the line defined by

        (x,y) = 1/2 (a+c,b+d) + t ((d-b)/(c-a),-1)    (t any real number)

and since the triangle is equilateral, we must have

        ||t ((d-b)/(c-a),-1)|| = sqrt(3)/2 ||(c,d)-(a,b)||

which yields t = +/- sqrt(3)/2 (c-a).  Thus the 3rd vertex is

        1/2 (a+c,b+d) +/- sqrt(3)/2 (d-b,a-c)

which must be irrational in at least one coordinate.