Answer: one triangle, with angles $100^\circ, 64^\circ, 16^\circ$

We want to find angles $x^\circ$, $y^\circ$ and $z^\circ$ so that $$x^2+y^2+z^2=180.$$

We know that the largest angle must be smaller than $180^\circ$ and bigger than $180^\circ/3 = 60^\circ$. So the largest angle must be $169^\circ, 144^\circ, 121^\circ, 100^\circ, 81^\circ$ or $64^\circ$.

largest angle	medium angle	smallest angle	all squares?
$169$	$9$	$2$	no
	$4$	$7$	no
$144$	$36$	$0$	not a triangle
	$25$	$11$	no
	$16$	$20$	no
$121$	$49$	$10$	no
	$36$	$23$	no
$100$	$64$	$16$	yes!
	$49$	$31$	no
	$36$	$44$	no
$81$	$81$	$18$	no
	$64$	$35$	no
	$49$	$50$	no
$64$	$64$	$52$	no
	$49$	$67$	no

So there is exactly one triangle with all three angles perfect squares, viz a triangle with angles $10^2,8^2,4^2$