DEFDBL A-Z
CLS
FOR a = 100 TO 998
  a2 = a * a: IF a2 > 99999 THEN EXIT FOR

s$ = LTRIM$(STR$(a2))
good = 1
FOR i = 1 TO 4
 FOR j = i + 1 TO 5
  IF MID$(s$, i, 1) = MID$(s$, j, 1) THEN good = 0: EXIT FOR
 NEXT
 IF good = 0 THEN EXIT FOR
NEXT

IF good THEN

FOR b = a + 1 TO 999
  b2 = b * b: IF b2 > 99999 THEN EXIT FOR
  c2 = a2 + b2: IF c2 > 99999 THEN EXIT FOR
  sr = INT(SQR(c2) + .5)
 
IF sr * sr = c2 THEN
          
  s$ = LTRIM$(STR$(b2))
  s1$ = LTRIM$(STR$(c2))
  good = 1
  FOR i = 1 TO 4
   FOR j = i + 1 TO 5
    IF MID$(s1$, i, 1) = MID$(s1$, j, 1) THEN good = 0: EXIT FOR
    IF MID$(s$, i, 1) = MID$(s$, j, 1) THEN good = 0: EXIT FOR
   NEXT
   IF good = 0 THEN EXIT FOR
  NEXT

  IF good THEN
     PRINT a2, b2, c2
     PRINT a, b, sr
     PRINT
  END IF

END IF

NEXT b

END IF

NEXT a

finds

18496         74529         93025
136           273           305

with the 5-digit squares on top and A, B and C below. Note that the program doesn't check that all 10 digits have been used, but the answer is still unique.