Produce n n-squares of integers with rotating *'s in consecuitive corners

Dyalog APL, 57 bytes

Requires ⎕IO←0 which is default on many systems.

Assuming double-spacing is permitted, as per the OP's first example.

{A←⊃∘⍕¨?10⍴⍨3⍴⍵⋄A[(⍳⍵),¨⍵⍴2↑¨1⍵(⍵ ⍵)(1⍵)-1]←'*'⋄⍉⎕FMT A}

TryAPL online!

Non-competing 49-byte solution (Dyalog APL 16.0):

{⍉⎕FMT'*'@((⍳⍵),¨⍵⍴2↑¨1⍵(⍵ ⍵)(1⍵)-1)⊃∘⍕¨?10⍴⍨3⍴⍵}

05AB1E, 50 49 bytes

3mF9Ý.R}J¹ô¹ävy`¦'*ìN4%©>ir}®iRr}®<iR})ˆ}¯øvyðý}»

Explanation

Examples for input = 4.

First we create a string of input^3 random numbers between 0 and 9.

3mF9Ý.R}J

producing

6799762549425893341317984133999075245812305412010122884262903656

Then we split that into pieces each the size of the input.
That is further split into input pieces.

¹ô¹ä

This gives us a matrix of numbers.

[['6799', '7625', '4942', '5893'], 
 ['3413', '1798', '4133', '9990'], 
 ['7524', '5812', '3054', '1201'], 
 ['0122', '8842', '6290', '3656']]

We then loop over the rows of the matrix, inserting asterisks in the right places.

v                          } # for each row in matrix 
 y`                          # flatten list to stack
   ¦'*ì                      # replace the first digit of the last number with "*"
       N4%©>ir}              # if row-nr % 4 == 0, move the number with "*" to the front
               ®iRr}         # if row-nr % 4 == 1, move the number with "*" to the front
                             # and reverse the number, moving "*" to the numbers right side
                    ®<iR}    # if row-nr % 4 == 2, reverse the number, moving "*" 
                             # to the numbers right side
                         )ˆ  # wrap row in a list and add to global array

Now we have the matrix with a "*" on each row, but we want an asterisk per column.

[['*893', '4942', '7625', '6799'], 
 ['099*', '4133', '1798', '3413'], 
 ['7524', '5812', '3054', '102*'], 
 ['0122', '8842', '6290', '*656']]

So we zip this list turning rows into columns and vice versa.

[['*893', '099*', '7524', '0122'], 
 ['4942', '4133', '5812', '8842'], 
 ['7625', '1798', '3054', '6290'], 
 ['6799', '3413', '102*', '*656']]

All that's left now is to format the output.

vyðý}»

Joining the rows on spaces and the columns on newlines gives us the final result.

*893 099* 7524 0122
4942 4133 5812 8842
7625 1798 3054 6290
6799 3413 102* *656

Try it online!

Old 50 byte solution

F¹Fõ¹F9Ý.R«}}¦'*ì})¹ävyN4%©>iR}®iíÁ}®<ií}})øvyðý}»

Java 7, 372 370 366 bytes

String c(int i){String s="*",a[][]=new String[i][i],t,r="";int j=0,k,z=i-1,m=(int)Math.pow(10,z);for(;j<i;j++)for(k=0;k<i;a[j][k++]=(new java.util.Random().nextInt((int)Math.pow(10,i)-m+1)+m)+"");for(j=0;j<i;k=j%4,t=a[m=k<2?0:z][k],a[m][j++]=k<1|k>2?s+t.substring(1,i):t.substring(0,z)+s);for(j=0;j<i;j++,r+="\n")for(k=0;k<i;r+=a[j][k++]+" ");return i<1?"":i<2?s:r;}

Can probably be golfed some more by placing the asterisks and creating the output at the same time, instead of one by one.

Ungolfed & test code:

Try it here.

class M{
  static String c(int i){
    String s = "*",
           a[][] = new String[i][i],
           t,
           r = "";
    int j = 0,
        k,
        z = i-1,
        m = (int)Math.pow(10, z);
    for(; j < i; j++){
      for(k = 0; k < i; a[j][k++] = (new java.util.Random().nextInt((int)Math.pow(10, i) - m + 1) + m)+"");
    }
    for(j = 0; j < i; k = j % 4,
                      t = a[m = k < 2 ? 0 : z][k],
                      a[m][j++] = k < 1 | k > 2
                                   ? s + t.substring(1, i)
                                   : t.substring(0, z) + s);

    for(j = 0; j < i; j++,
                      r += "\n"){
      for(k = 0; k < i; r += a[j][k++] + " ");
    }
    return i < 1
            ? ""
            : i < 2
               ? s
               : r;
  }

  public static void main(String[] a){
    for (int i = 0; i < 6; i++) {
      System.out.println(c(i));
      System.out.println();
    }
  }
}

Possible output:

(empty)

*

*9 4* 
92 47 


*25 55* 754 
910 197 108 
635 439 35* 


*512 407* 9646 5017 
1663 3847 9772 3149 
7796 2997 5494 1362 
7283 9720 242* *539 


*0726 7743* 52096 50958 *0726 
60322 20914 76387 92716 41579 
89994 18781 33379 84189 31777 
11781 89323 12180 51814 63536 
58411 32935 5168* *6597 43216