Your question does not provide enough information to fully respond, because it is unclear what the context of the representation would be. However, for purposes of litigation, New York State's Civil Practice Law and Rules section 321(a) requires that corporations who prosecute or defend a civil action shall be represented by an attorney. A single member LLC falls within the scope of that requirement. If your LLC is involved in, or contemplating involvement in civil litigation, you should...
If you proceed with filing a claim in small claims court, it is important to follow the rules. The person that you are suing has a right to be notified of the lawsuit, and you will need to at least have a mailing address for the defendant to proceed. The New York State Courts have a useful handbook that you can refer to for help, and it may answer other questions you may have about the process. It is located at: http://www.courts.state.ny.us/courts/townandvillage/pdfs/SmallClaimsHandbook.pdf
In New York State, employment is generally "at will." This means that, unless you have a contract that provides otherwise, an employee can quit at any time, and an employer can fire an employee at any time. However, an employer may not fire an employee based upon criteria that is illegal for consideration, such as race, gender, and religion, among other considerations.
It is always a mistake to ignore notice of a lawsuit against you, regardless of whether you believe that there is any merit to the claims in the documents that you received. If you were properly served with notice of the lawsuit, if you do not respond, it may be possible for the plaintiff to obtain a judgment against you. Receipt of service of a lawsuit should be taken seriously, and if you do not know how to respond, you should seek legal counsel to preserve your rights.