Here is my version. It's about 50 bytes larger, but averages 1889.75cc. It does use stack and shadow registers, but it could be made faster by totally unrolling (which would be about 300 bytes of code).
sqrt32:
;Input: HLDE
;Output: DE is the square root, AHL is the remainder
;Destroys: D'E', H'L'
;Speed: 248+{0,44}+3*sqrt32sub+sqrt32sub_2+sqrt32_iter15
;min: 1697cc
;max: 2086cc
;avg: 1889.75cc
;
;Python implementation:
;  remainder = 0
;  acc = 0
;  for k in range(16):
;    acc<<=1
;    x&=0xFFFFFFFF
;    x<<=2
;    y=x>>32
;    remainder<<=2
;    remainder+=y
;    if remainder>=acc*2+1:
;      remainder-=(acc*2+1)
;      acc+=1
;  return [acc,remainder]
;
  di
  exx 
  ld hl,0         ;remainder
  ld d,h \ ld e,h ;acc
  exx

  ld a,h \ call sqrt32sub \ exx
  ld a,l \ call sqrt32sub \ exx
  ld a,d \ call sqrt32sub \ exx
;Now we have four more iterations
;The first two are no problem
  ld a,e
  exx
  call sqrt32sub_2

;On the next iteration, HL might temporarily overflow by 1 bit
  call sqrt32_iter15

;On the next iteration, HL is allowed to overflow, DE could overflow with our current routine, but it needs to be shifted right at the end, anyways
sqrt32_iter16:
  add a,a
  adc hl,hl
  rla
  adc hl,hl
  rla
;AHL - (DE+DE+1)
  sbc hl,de \ sbc a,0
  inc e
  sbc hl,de \ sbc a,0
  ret p
  add hl,de
  adc a,0
  dec e
  add hl,de
  adc a,0
  ret


sqrt32sub:
;min: 391cc
;max: 483cc
;avg: 437cc
  exx
  call sqrt32sub_2

sqrt32sub_2:
;min: 185cc
;max: 231cc
;avg: 208cc
  call +_

_:
;min: 84cc
;max: 107cc
;avg: 95.5cc

  sll e \ rl d      ;sla e \ rl d \ inc e

  add a,a
  adc hl,hl
  add a,a
  adc hl,hl

  sbc hl,de
  inc e
  ret nc
  dec e
  add hl,de
  dec e
  ret

sqrt32_iter15:
;91+{8,0+{0,23}}
;min: 91cc
;max: 114cc
;avg: 100.75cc

  sll e \ rl d      ;sla e \ rl d \ inc e
  add a,a
  adc hl,hl
  add a,a
  adc hl,hl       ;This might overflow!
  jr c,sqrt32_iter15_br0
;
  sbc hl,de
  inc e
  ret nc
  dec e
  add hl,de
  dec e
  ret
sqrt32_iter15_br0:
  or a
  sbc hl,de
  inc e
  ret

EDIT:
Oh jeez, here is an even bigger version that uses less stack space and doesn't use shadow registers or index registers:
sqrt32:
;Input: HLDE
;speed: 238+{0,1}+{0,44}+sqrtHL+3*sqrt32sub_2+sqrt32_iter15
;min: 1260
;max: 1506
;avg: 1377.75

  push de
  call sqrtHL
  pop bc
  add a,a
  ld e,a
  jr nc,+_
  inc d
_:

  ld a,b
  call sqrt32sub_2
  call sqrt32sub_2
;Now we have four more iterations
;The first two are no problem
  ld a,c
  call sqrt32sub_2

;On the next iteration, HL might temporarily overflow by 1 bit
  call sqrt32_iter15

;On the next iteration, HL is allowed to overflow, DE could overflow with our current routine, but it needs to be shifted right at the end, anyways
sqrt32_iter16:
  add a,a
  adc hl,hl
  rla
  adc hl,hl
  rla
;AHL - (DE+DE+1)
  sbc hl,de \ sbc a,0
  inc e
  or a
  sbc hl,de \ sbc a,0
  ret p
  add hl,de
  adc a,0
  dec e
  add hl,de
  adc a,0
  ret

sqrt32sub_2:
;min: 185cc
;max: 231cc
;avg: 208cc
  call +_

_:
;min: 84cc
;max: 107cc
;avg: 95.5cc

  sll e \ rl d
  add a,a \ adc hl,hl
  add a,a \ adc hl,hl

  sbc hl,de
  inc e
  ret nc
  dec e
  add hl,de
  dec e
  ret

sqrt32_iter15:
;91+{8,0+{0,23}}
;min: 91cc
;max: 114cc
;avg: 100.75cc

  sll e \ rl d      ;sla e \ rl d \ inc e
  add a,a
  adc hl,hl
  add a,a
  adc hl,hl       ;This might overflow!
  jr c,sqrt32_iter15_br0
;
  sbc hl,de
  inc e
  ret nc
  dec e
  add hl,de
  dec e
  ret
sqrt32_iter15_br0:
  or a
  sbc hl,de
  inc e
  ret
.echo $-sqrt32

sqrtHL:
;returns A as the sqrt, HL as the remainder, D = 0
;min: 376cc
;max: 416cc
;avg: 393cc
  ld de,$5040
  ld a,h
  sub e
  jr nc,+_
  add a,e
  ld d,$10
_:
  sub d
  jr nc,+_
  add a,d
  .db $01   ;start of ld bc,** which is 10cc to skip the next two bytes.
_:
  set 5,d
  res 4,d
  srl d

  set 2,d
  sub d
  jr nc,+_
  add a,d
  .db $01   ;start of ld bc,** which is 10cc to skip the next two bytes.
_:
  set 3,d
  res 2,d
  srl d

  inc d
  sub d
  jr nc,+_
  add a,d
  dec d   ;this resets the low bit of D, so `srl d` resets carry.
  .db $06   ;start of ld b,* which is 7cc to skip the next byte.
_:
  inc d
  srl d
  ld h,a


  sbc hl,de
  ld a,e
  jr nc,+_
  add hl,de
_:
  ccf
  rra
  srl d
  rra
  ld e,a

  sbc hl,de
  jr nc,+_
  add hl,de
  .db $01   ;start of ld bc,** which is 10cc to skip the next two bytes.
_:
  or %00100000
  xor %00011000
  srl d
  rra
  ld e,a


  sbc hl,de
  jr nc,+_
  add hl,de
  .db $01   ;start of ld bc,** which is 10cc to skip the next two bytes.
_:
  or %00001000
  xor %00000110
  srl d
  rra
  ld e,a
  sbc hl,de
  jr nc,+_
  add hl,de
  srl d
  rra
  ret
_:
  inc a
  srl d
  rra
  ret
.echo $-sqrtHL
It does use the 16-bit square root routine here to take care of the first 16 bits Combined, it is 194 bytes.
EDIT2: Forgot that sqrtHL didn't preserve BC, fixed that. Now it seems the last bit of the remainder might be broken, so I have to fix that
EDIT3: Fixed the bug in the bottom bit I just needed to reset the carry flag before the second subtraction in the final iteration.

---

EDIT4: In a scenario where you don't have RAM for a stack, we can hardcode it! It even saves 54cc (but adds 20 bytes). 10cc of that 54cc is just due to not having an ending RET. I also switched input to HLIX instead of HLDE.
I reorganized the code so that it would be "obvious" that sqrt32 is an in-line routine, so I put it at the end (in practice, the subroutines would probably be toward the end of mem). The precomputed stack is inserted just before sqrt32.
sqrt32sub_2:
;min: 178cc
;max: 224cc
;avg: 201cc
  jp return4
return4:
;min: 84cc
;max: 107cc
;avg: 95.5cc

  sll e \ rl d
  add a,a \ adc hl,hl
  add a,a \ adc hl,hl

  sbc hl,de
  inc e
  ret nc
  dec e
  add hl,de
  dec e
  ret

sqrt32_iter15:
;91+{8,0+{0,23}}
;min: 91cc
;max: 114cc
;avg: 100.75cc

  sll e \ rl d      ;sla e \ rl d \ inc e
  add a,a
  adc hl,hl
  add a,a
  adc hl,hl       ;This might overflow!
  jr c,sqrt32_iter15_br0
;
  sbc hl,de
  inc e
  ret nc
  dec e
  add hl,de
  dec e
  ret
sqrt32_iter15_br0:
  or a
  sbc hl,de
  inc e
  ret

sqrtHL:
;returns A as the sqrt, HL as the remainder, D = 0
;min: 376cc
;max: 416cc
;avg: 393cc
  ld de,$5040
  ld a,h
  sub e
  jr nc,+_
  add a,e
  ld d,$10
_:
  sub d
  jr nc,+_
  add a,d
  .db $01   ;start of ld bc,** which is 10cc to skip the next two bytes.
_:
  set 5,d
  res 4,d
  srl d

  set 2,d
  sub d
  jr nc,+_
  add a,d
  .db $01   ;start of ld bc,** which is 10cc to skip the next two bytes.
_:
  set 3,d
  res 2,d
  srl d

  inc d
  sub d
  jr nc,+_
  add a,d
  dec d   ;this resets the low bit of D, so `srl d` resets carry.
  .db $06   ;start of ld b,* which is 7cc to skip the next byte.
_:
  inc d
  srl d
  ld h,a


  sbc hl,de
  ld a,e
  jr nc,+_
  add hl,de
_:
  ccf
  rra
  srl d
  rra
  ld e,a

  sbc hl,de
  jr nc,+_
  add hl,de
  .db $01   ;start of ld bc,** which is 10cc to skip the next two bytes.
_:
  or %00100000
  xor %00011000
  srl d
  rra
  ld e,a


  sbc hl,de
  jr nc,+_
  add hl,de
  .db $01   ;start of ld bc,** which is 10cc to skip the next two bytes.
_:
  or %00001000
  xor %00000110
  srl d
  rra
  ld e,a
  sbc hl,de
  jr nc,+_
  add hl,de
  srl d
  rra
  ret
_:
  inc a
  srl d
  rra
  ret
sqrt32_stack:
.dw return0
.dw return4   ;subroutine
.dw return1
.dw return4   ;subroutine
.dw return2
.dw return4   ;subroutine
.dw return3
.dw return5
sqrt32_stack_end:




sqrt32:
;Input: HLIX
;Output: DE is the sqrt, AHL is the remainder
;min: 1203
;max: 1455
;avg: 1323.75
  ld sp,sqrt32_stack
  jp sqrtHL
return0:
  add a,a
  ld e,a
  jr nc,+_
  inc d
_:

  ld a,ixh
  jp sqrt32sub_2
return1:
  jp sqrt32sub_2
return2:
;Now we have four more iterations
;The first two are no problem
  ld a,ixl
  jp sqrt32sub_2
return3:
;On the next iteration, HL might temporarily overflow by 1 bit
  jp sqrt32_iter15
return5:

;On the next iteration, HL is allowed to overflow, DE could overflow with our current routine, but it needs to be shifted right at the end, anyways
sqrt32_iter16:
  add a,a
  adc hl,hl
  rla
  adc hl,hl
  rla
;AHL - (DE+DE+1)
  sbc hl,de \ sbc a,0
  inc e
  or a
  sbc hl,de \ sbc a,0
  jp p,+_
  add hl,de
  adc a,0
  dec e
  add hl,de
  adc a,0
_:
  ;...


I bet if the Z80 gods got in here they could do even better! As it is, I see a neat optimization saving 3 bytes and on average 5cc, but I think I'll wait to make more edits. Basically, in the last iteration instead of:
sqrt32_iter16:
  add a,a
  adc hl,hl
  rla
  adc hl,hl
  rla
;AHL - (DE+DE+1)
  sbc hl,de \ sbc a,0
  inc e
  or a
  sbc hl,de \ sbc a,0
  ret p
  add hl,de
  adc a,0
  dec e
  add hl,de
  adc a,0
  ret
It could be:
sqrt32_iter16:
  add a,a
  ld b,a  ;either 0x00 or 0x128
  adc hl,hl
  rla
  adc hl,hl
  rla
;AHL - (DE+DE+1)
  sbc hl,de \ sbc a,b
  inc e
  or a
  sbc hl,de \ sbc a,b
  ret p
  add hl,de
  adc a,b
  dec e
  add hl,de
  adc a,b
  ret

I'll have to test that and other optimizations after work (if I have time tonight).

EDIT:
Here is an unrolled version that includes the above optimization and works with input=HLIX. This code is 167 bytes, but including the sqrtHL routine posted earlier, the total size is 266 bytes.
sqrtHLIX:
;Input: HLIX
;Output: DE is the sqrt, AHL is the remainder
;speed: 754+{0,1}+6{0,6}+{0,3+{0,18}}+{0,38}+sqrtHL
;min: 1130
;max: 1266
;avg: 1190.5
;167 bytes

  call sqrtHL
  add a,a
  ld e,a
  jr nc,+_
  inc d
_:

  ld a,ixh
  sll e \ rl d
  add a,a \ adc hl,hl
  add a,a \ adc hl,hl
  sbc hl,de
  jr nc,+_
  add hl,de
  dec e
  .db $FE     ;start of `cp *`
_:
  inc e

  sll e \ rl d
  add a,a \ adc hl,hl
  add a,a \ adc hl,hl
  sbc hl,de
  jr nc,+_
  add hl,de
  dec e
  .db $FE     ;start of `cp *`
_:
  inc e

  sll e \ rl d
  add a,a \ adc hl,hl
  add a,a \ adc hl,hl
  sbc hl,de
  jr nc,+_
  add hl,de
  dec e
  .db $FE     ;start of `cp *`
_:
  inc e

  sll e \ rl d
  add a,a \ adc hl,hl
  add a,a \ adc hl,hl
  sbc hl,de
  jr nc,+_
  add hl,de
  dec e
  .db $FE     ;start of `cp *`
_:
  inc e

;Now we have four more iterations
;The first two are no problem
  ld a,ixl
  sll e \ rl d
  add a,a \ adc hl,hl
  add a,a \ adc hl,hl
  sbc hl,de
  jr nc,+_
  add hl,de
  dec e
  .db $FE     ;start of `cp *`
_:
  inc e

  sll e \ rl d
  add a,a \ adc hl,hl
  add a,a \ adc hl,hl
  sbc hl,de
  jr nc,+_
  add hl,de
  dec e
  .db $FE     ;start of `cp *`
_:
  inc e

sqrt32_iter15:
;On the next iteration, HL might temporarily overflow by 1 bit
  sll e \ rl d      ;sla e \ rl d \ inc e
  add a,a
  adc hl,hl
  add a,a
  adc hl,hl       ;This might overflow!
  jr c,sqrt32_iter15_br0
;
  sbc hl,de
  jr nc,+_
  add hl,de
  dec e
  jr sqrt32_iter16
sqrt32_iter15_br0:
  or a
  sbc hl,de
_:
  inc e

;On the next iteration, HL is allowed to overflow, DE could overflow with our current routine, but it needs to be shifted right at the end, anyways
sqrt32_iter16:
  add a,a
  ld b,a        ;either 0x00 or 0x80
  adc hl,hl
  rla
  adc hl,hl
  rla
;AHL - (DE+DE+1)
  sbc hl,de \ sbc a,b
  inc e
  or a
  sbc hl,de \ sbc a,b
  ret p
  add hl,de
  adc a,b
  dec e
  add hl,de
  adc a,b
  ret

Thank you so much for the inspiration to work on this routine! I was able to take the results here and make my float routines a little bit faster and I saved 222 bytes in all!