5   A=100:open "8primes.txt" for output as #2
   10   while A<1000
   20      A=nxtprm(A)
   30      if A<1000 and (A\10)@10>0 then
   35         :B=100
   40         :while B<1000
   50              :B=nxtprm(B)
   60              :if B<1000 then
   70                     :C=100
   80                     :while C<1000
   90                          :C=nxtprm(C)
  100                          :Av=100*(A\100)+10*(B\100)+C\100
  110                          :Bv=100*((A\10)@10)+10*((B\10)@10)+((C\10)@10)
  130                          :Cv=100*(A@10)+10*(B@10)+C@10
  140                          :D1=100*(A\100)+10*((B\10)@10)+C@10
  150                          :D2=100*(A@10)+10*((B\10)@10)+C\100
  160                          :if prmdiv(Av)=Av and prmdiv(Bv)=Bv and prmdiv(Cv)=Cv then
  170                              :if prmdiv(D1)=D1 and prmdiv(D2)=D2 then
  180                                  :print #2,A;B;C,Av;Bv;Cv
  190                                  :inc Solct
  200                              :endif
  210                          :endif
  220                     :wend
  230              :endif
  240         :wend
  250      :endif
  260   wend
  270   print Solct

finds 6825 ways of arranging it so that all 8 specified lines contain 3-digit primes, not necessarily unique.

If all 8 primes are different, you cannot use just two different digits repeated to form the square. The most different primes present is five for this purpose:

311 
113 
331

313 
113 
131

313 
311 
131

313 
313 
131

331 
113 
311

331 
113 
331

911 
199 
911

911 
199 
991

Symmetry about the diagonals is possible replicating just three unique digits, and in a couple of cases, just two:

Three unique digits:

113
151
311

113
181
311

131
353
131

131
383
131

151
599
191

151
599
199

181
883
131

191
991
113

191
997
179

193
911
313

197
991
719

211
131
113

223
211
311

227
223
733

227
227
773

227
229
797

229
211
911

229
227
977

229
229
997

233
311
311

233
313
331

233
373
337

277
727
773

311
139
191

311
151
113

311
181
113

Two unique digits:

313
113
331

911
199
191

Symmetry about the other diagonal (some about both diagonals; all with 3 unique digits repeated as necessary):

113
151
311

113
181
311

131
353
131

131
383
131

191
199
311

311
151
113

311
181
113

313
151
313

313
181
313

313
311
733

337
353
733

337
383
733

373
757
373

373
787
373

733
353
337

733
383
337

797
929
797

Third criterion:

Some of the below use fewer than five different digits, but those with five different digits are included as well and are marked with an *:

223
211
311

223    *
241
911

223
251
311

223    *
251
911

223
271
311

223    *
271
911

223
281
311

223    *
281
911

227
223
733

227
283
733

227
293
733

229
211
911

229
227
977

229    *
251
311

229    *
281
311

331
359
199

331
389
199

449
421
911

557
523
733

557
593
733

661
659
199

773
751
311

881
809
199

881
839
199

881
859
199

883
811
311

883
881
311

887
823
733

887
853
733

887
883
733

997
953
733

997
983
733

 

Posted by Charlie on 2013-10-09 18:06:22