Let the rectangular box sides be a,b and c. Call integers x,y a pythagorean double if $x^2 + y^2 = z^2$ for some integer z.
Clearly a,b and c must be integers; that's three paths.
Also, any two of them must form a pythagorean double so that the ant can get from one corner of a face to the opposite corner. That means a,b and c must be distinct.
Finally $(a+b),c$ must form a pythagorean double (where c is the longest side), so that the ant can get from one corner to the opposite corner of the box.
So far so good. This means c must form a pythagorean double with a, with b, and with a+b; and a,b must form a pythagorean double.
One such set of numbers is:
a = 44, b = 117, c = 240 - these are the ratios of the sides of the box.
These give pythagorean triples of 44,117,125; 44,240,244; 117,240,267; 161,240,289
So the other four path lengths are (some multiples of) 125, 244, 267 and 289.
There may be others.