The 3x3 Hexa Prime Square Puzzle

05AB1E, 23 21 bytes

Uses CP-1252 encoding.

œJvy3ôD€SøJìHDps2›*P—

Too slow for TIO.

Explanation

œJ                     # all permutations of input as strings
  v                    # for each permutation
                       # EXAMPLE: 2E3815DD5
   y3ô                 # split in pieces of 3
                       # EXAMPLE: ['2E3','815','DD5']
      D                # duplicate
                       # EXAMPLE: ['2E3','815','DD5'], ['2E3','815','DD5']
       €SøJ            # zip the copy to swap rows and columns 
                       # EXAMPLE: ['2E3','815','DD5'], ['28D','E1D','355']
           ì           # attach them to the same list
                       # EXAMPLE: ['2E3','815','DD5','28D','E1D','355']
            H          # convert from base 16 to base 10
                       # EXAMPLE: [739, 2069, 3541, 653, 3613, 853]
             D         # duplicate
                       # EXAMPLE: [739, 2069, 3541, 653, 3613, 853],[739, 2069, 3541, 653, 3613, 853]
              p        # check first copy for primality
                       # EXAMPLE: [739, 2069, 3541, 653, 3613, 853],[1,1,1,1,1,1]
               s2›     # check that each in second copy is larger than 2
                       # EXAMPLE: [1,1,1,1,1,1],[1,1,1,1,1,1]
                  *    # pairwise multiplication
                       # EXAMPLE: [1,1,1,1,1,1]
                   P   # product (1 if all were primes larger than 2, else 0)
                       # EXAMPLE: 1
                    —  # if 1, print y
                       # EXAMPLE: 2E3815DD5

Python 2, 212 206 197 194 bytes

Requires input enclosed in quotes, like "123558dde"

from itertools import*
k,p,P=3,4,[]
while k<5e3:P+=[k][:p%k];p*=k;k+=1
print[s for s in map(''.join,permutations(input()))if all(int(s[3*i:][:3],16)in P and int(s[i::3],16)in P for i in(0,1,2))]

Saving 9 and 3 bytes thanks to Jonathan Allan

Found new prime filter from xnor (modified the square away, since we dont want 2 as prime here), old prime filter is from Bob

Jelly, 34 30 bytes

i@€ØH’
s3µ;ZÇ€ḅ⁴µ>2aÆPẠ
Œ!ÇÐfḢ

(I should be able to use an nfind to just fetch the first match, 1# in place of ÐfḢ, for less bytes and more speed, but I'm seeing errors when I try. EDIT: wrote some changes to possibly implement this in Jelly.)

Brute force search of all permutations, filtered for the criteria, returning first match.
Way too slow for TtyItOnline. Local output examples:

C:\Jelly\jelly-master>python jelly -fu test.txt "123558DDE"
28DE1D355
C:\Jelly\jelly-master>python jelly -fu test.txt "1155578AB"
11B8A5557

How?

i@€ØH’ - Link 1, convert from hexadecimal string to integer list: string
   ØH  - yield hexadecimal characters, "0123456789ABCDEF"
i@€    - index of €ach character of s in hex chars
     ’ - decrement (vectorises) (from 1 based jelly index to place value)

s3µ;ZÇ€ḅ⁴µ>2aÆPẠ - Link 2, check if a flattened square is "all prime": string
s3               - split into threes (rows)
  µ              - monadic chain separation
   ;             - concatenate with
    Z            - transpose (columns)
     Ç€          - call last link (1) as a monad for €ach string in the list
                       -> list of integer lists
       ḅ⁴        - convert from base 16 (vectorises) -> list of decimals
         µ       - monadic chain separation
          >2     - greater than 2
            a    - and
             ÆP  - isPrime? -> list of 1s and 0s
               Ạ - all truthy?

Œ!ÇÐfḢ  - Main link: string
Œ!      - all permutations of the string
   Ðf   - filter keeping entries that evaluate to truthily for
  Ç     - last link (3) as a monad
     Ḣ  - head - return first entry