Quit Whining; Start Quining
GolfScript, 16 bytes

{%}+")1$~":x`+0x


Try it online in Web GolfScript.

Example run

$ echo -n 3 | golfscript quining.gs      
{3 % ")1$~"}0)1$~
$ echo -n 3 | golfscript quining.gs | golfscript 
{3 % ")1$~"}1)1$~
$ echo -n 3 | golfscript quining.gs | golfscript | golfscript 
{3 % ")1$~"}2)1$~
$ echo -n 3 | golfscript quining.gs | golfscript | golfscript | golfscript 
{3 % ")1$~"}0)1$~


Generator

                 # (implicit) Read from STDIN.
{%}+             # Concatenate the read input with the block.
                 # For example, "3"{%}+ -&gt; {3 %}.
    ")1$~":x     # Push that string and save it in the the variable `x'.
            `+   # Inspect the string and concatenate it with the block.
                 # For example, {3 %}'")1$~"'+ -&gt; {3 % ")1$~"}
              0x # Push a 0 and the string stored in `x'.


Modular quine

For example, the modular quine {3 % ")1$~"}0)1$~ works as follows.

{          }      # Define a block.
 3 %              # Take the integer on the stack modulo 3.
     ")1$~"       # Push that string.
            0)    # Push 0 and increment. Pushes 1.
              1$~ # Copy the block from the bottom of the stack and execute it.


The original copy of the block, the modified integer and the string the block pushed are printed, resulting in {3 % ")1$~"}1)1$~.

All printed programs have a trailing linefeed, which GolfScript prints automatically.