Does the Paillier system remain secure if it is used to encrypt only binary values, i.e. {0, 1}? Is there any security compromises if the Paillier system was used to encrypt only binary message in {0, 1}? i.e., plaintexts are either 0 or 1.
No, there are no security compromises; the Pallier system remains secure.  Both messages are in the supported message space and as Paillier encryption provides IND-CPA security you are safe when doing this.