C, 67 bytes

i,n;main(p){for(scanf("%d",&i),n=i;--i;p=p*i*i%n);putchar(48+p%n);}

Prints !1 (a falsey value, by Peter Taylor's definition) 0 if (n-1)!^2 == 0 (mod n), and 1 otherwise.

EDIT: After some discussion in chat, puts("!1"+p%n) seems to be considered a bit cheaty, so I've replaced it. The result is one byte longer.

EDIT: Fixed for big inputs.

56 bytes (unary input)

As recommended in the comments by pawel.boczarski, I could take input in unary by reading the number of command line arguments:

p=1,n;main(i){for(n=--i;--i;p=p*i*i%n);putchar(48+p%n);}

invoking the program like

$ ./a.out 1 1 1 1 1
1                        <-- as 5 is prime

C, 67 bytes

i,n;main(p){for(scanf("%d",&i),n=i;--i;p=p*i*i%n);putchar(48+p%n);}

Prints !1 (a falsey value, by Peter Taylor's definition) 0 if (n-1)!^2 == 0 (mod n), and 1 otherwise.

EDIT: After some discussion in chat, puts("!1"+p%n) seems to be considered a bit cheaty, so I've replaced it. The result is one byte longer.

EDIT: Fixed for big inputs.

Shorter solutions

56 bytes: As recommended in the comments by pawel.boczarski, I could take input in unary by reading the number of command line arguments:

p=1,n;main(i){for(n=--i;--i;p=p*i*i%n);putchar(48+p%n);}

invoking the program like

$ ./a.out 1 1 1 1 1
1                        <-- as 5 is prime

51 bytes: If you allow "output" by means of return codes:

p=1,n;main(i){for(n=--i;--i;p=p*i*i%n);return p%n;}

C, 63 bytes

i,n;main(p){for(scanf("%d",&i),n=i;--i;p*=i*i);putchar(48+p%n);}

Prints !1 (a falsey value, by Peter Taylor's definition) 0 if (n-1)!^2 == 0 (mod n), and 1 otherwise.

EDIT: After some discussion in chat, puts("!1"+p%n) seems to be considered a bit cheaty, so I've replaced it. The result is one byte longer.

C, 67 bytes

i,n;main(p){for(scanf("%d",&i),n=i;--i;p=p*i*i%n);putchar(48+p%n);}

Prints !1 (a falsey value, by Peter Taylor's definition) 0 if (n-1)!^2 == 0 (mod n), and 1 otherwise.

EDIT: After some discussion in chat, puts("!1"+p%n) seems to be considered a bit cheaty, so I've replaced it. The result is one byte longer.

EDIT: Fixed for big inputs.

56 bytes (unary input)

As recommended in the comments by pawel.boczarski, I could take input in unary by reading the number of command line arguments:

p=1,n;main(i){for(n=--i;--i;p=p*i*i%n);putchar(48+p%n);}

invoking the program like

$ ./a.out 1 1 1 1 1
1                        <-- as 5 is prime

C, 63 bytes

i,n;main(p){for(scanf("%d",&i),n=i;--i;p*=i*i);puts("!1"+p%n);}

Prints !1 (a falsey value, by Peter Taylor's definition) if (n-1)!^2 == 0 (mod n), and 1 otherwise.

C, 63 bytes

i,n;main(p){for(scanf("%d",&i),n=i;--i;p*=i*i);putchar(48+p%n);}

Prints !1 (a falsey value, by Peter Taylor's definition) 0 if (n-1)!^2 == 0 (mod n), and 1 otherwise.

EDIT: After some discussion in chat, puts("!1"+p%n) seems to be considered a bit cheaty, so I've replaced it. The result is one byte longer.